What’s the difference between uniform and coarse structures? (I)

See this, if you need: uniform structure and coarse structure.

Given a set $X$, endowed with an uniform structure or with a coarse structure, Instead of working with subsets of $X^{2}$, I prefer to think about the trivial groupoid $X \times X$.

Groupoids. Recall that a groupoid is a small category with all arrows invertible. We may define a groupoid in term of it’s collection of arrows, as follows: a set $G$ endowed with a partially defined operation $m: G^{(2)} \subset G \times G \rightarrow G$, $m(g,h) = gh$, and an inverse operation $inv: G \rightarrow G$, $inv(g) = g^{-1}$, satisfying some well known properties (associativity, inverse, identity).

The set $Ob(G)$ is the set of objects of the groupoid, $\alpha , \omega: G \rightarrow Ob(G)$ are the source and target functions, defined as:

– the source fonction is $\alpha(g) = g^{-1} g$,

– the target function is $\omega(g) = g g^{-1}$,

– the set of objects is $Ob(G)$ consists of all elements of $G$ with the form $g g^{-1}$.

Trivial groupoid. An example of a groupoid is $G = X \times X$, the trivial groupoid of the set $X$. The operations , source, target and objects are

$(x,y) (y,z) = (x,z)$, $(x,y)^{-1} = (y,x)$

$\alpha(x,y) = (y,y)$, $\omega(x,y) = (x,x)$,

– objects $(x,x)$ for all $x \in X$.

In terms of groupoids, here is the definition of an uniform structure.

Definition 1. (groupoid with uniform structure) Let $G$ be a groupoid. An uniformity $\Phi$ on $G$ is a set $\Phi \subset 2^{G}$ such that:

1. for any $U \in \Phi$ we have $Ob(G) \subset U$,

2. if $U \in \Phi$ and $U \subset V \subset G$ then $V \in \Phi$,

3. if $U,V \in \Phi$ then $U \cap V \in \Phi$,

4. for any $U \in \Phi$ there is $V \in \Phi$ such that $VV \subset U$,  where $VV$ is the set of all $gh$ with $g, h \in V$ and moreover $(g,h) \in G^{(2)}$,

5. if $U \in \Phi$ then $U^{-1} \in \Phi$.

Here is a definition for a coarse structure on a groupoid (adapted from the usual one, seen on the trivial groupoid).

Definition 2.(groupoid with coarse structure) Let $G$ be a groupoid. A coarse structure  $\chi$ on $G$ is a set $\chi \subset 2^{G}$ such that:

1′. $Ob(G) \in \chi$,

2′. if $U \in \chi$ and $V \subset U$ then $V \in \chi$,

3′. if $U,V \in \chi$ then $U \cup V \in \chi$,

4′. for any $U \in \chi$ there is $V \in \chi$ such that $UU \subset V$,  where $UU$ is the set of all $gh$ with $g, h \in U$ and moreover $(g,h) \in G^{(2)}$,

5′. if $U \in \chi$ then $U^{-1} \in \chi$.

Look pretty much the same, right? But they are not, we shall see the difference when we take into consideration how we generate such structures.

Question. Have you seen before such structures defined like this, on groupoids? It seems trivial to do so, but I cannot find this anywhere (maybe because I am ignorant).

UPDATE (01.08.2012): Just found this interesting paper  Coarse structures on groups, by Andrew Nicas and David Rosenthal, where they already did on groups something related to what I want to explain on groupoids. I shall be back on this.

Approximate groupoids may be useful

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I appreciated yesterday the talk of Harald Helfgott, here is the link to the arxiv paper which was the subject of his communication: On the diameter of permutation groups (with Á. Seress).

At the end of his talk Harald stated as a conclusion that (if I well understood) the approximate groups field should be regarded as the study of

“stable configurations under the action of a group”

where “stable configuration” means a set (or a family…) with some controlled behaviour of its growth under the “action of a group”, understood maybe in a lax, approximate sense.

This applies to approximate groups, he said, where the action is by left translations, but it could apply to the conjugate action as well, and evenin  more general settings, where one has a space where a group acts.

My understanding of this bold claim is that Harald suggests that the approximate groups theory is a kind of geometry in the sense of Felix Klein: the (approximate, in this case) study of stable configuratiuons under the action of a group.

Today I just remembered a comment that I have made last november on the blog of Tao, where I proposed to study “approximate groupoids”.

Why is this related?

Because a group acting on a set is just a particular type of groupoid, an action groupoid.

Here is the comment (link), reproduced further.

“Question (please erase if not appropriate): as a metric space is just a particular example of a normed groupoid, could you point me to papers where “approximate groupoids” are studied? For starters, something like an extension of your paper “Product set estimates for non-commutative groups”, along the lines in the introduction “one also consider partial sum sets …”, could be relevant, but I am unable to locate any. Following your proofs could be straightforward, but lengthy. Has this been done?

For example, a $k$  approximate groupoid $A \subset G$ of a groupoid  $G$ (where $G$ denotes the set of arrows) could be a

(i)- symmetric subset of $G$: $A = A^{-1}$, for any $x \in Ob(G)$  $id_{x} \in A$  and

(ii)- there is another, symmetric subset $K \subset G$ such that for any   $(u,v) \in (A \times A) \cap G^{(2)}$  there are $(w,g) \in (A \times K) \cap G^{(2)}$  such that $uv = wg$,

(iii)- for any $x \in Ob(G)$  we have $\mid K_{x} \mid \leq k$.

One may replace the cardinal, as you did, by a measure (or random walk kernel, etc), or even by a norm.”

UPDATE (28.10.2012): Apparently unaware about the classical proof of Ronald Brown,  by way of groupoids, of the Van Kampen theorem on the fundamental group of a union of spaces, Terence Tao has a post about this subject. I wonder if he is after some applications of his results on approximate groups in the realm of groupoids.

Gromov on entropy and Souriau

Gromov just posted on his page the paper In a Search for a Structure, Part 1: On Entropy. June 19, 2012.   With much interest I started to read it and my first impression is that I have seen something like this before (but I might be wrong, please excuse me if so) in the HUGE last paper (book)

Grammaire de la nature (version du 8 juillet 2007)

by Jean-Marie Souriau, the inventor of symplectic geometry and geometric quantization, among others.

Specifically, I refer to  the way Souriau treats probabilities in “CLE 8: Calcul des Hasards”, p. 209.

The book is a must-read! It is a summum of Souriau mathematical view of Nature, specifically concerning symmetry, entropy, relativity and quantum mechanics.

Gromov, with his enormous geometrical knowledge (different than Souriau’ though)  points to sofic groups, this  I need a lot of time to understand.

UPDATE: I am starting to understand the sofic group notion of Gromov and learning to appreciate it, it’s related to constructions with approximate groups, apparently.

Intrinsic characterizations of riemannian and sub-riemannian spaces (I)

In this post I explain what is the problem of intrinsic characterization of riemannian manifolds, in what sense has been solved in full generality by Nikolaev, then I shall comment on the proof of the Hilbert’s fifth problem by Tao.

In the next post there will be then some comments about Gromov’s problem of giving an intrinsic characterization of sub-riemannian manifolds, in what sense I solved this problem by adding a bit of algebra to it. Finally, I shall return to the characterization of riemannian manifolds, seen as particular sub-riemannian manifolds, and comment on the differences between this characterization and Nikolaev’ one.

1. History of the problem for riemannian manifolds. The problem of giving an intrinsic characterization of riemannian manifolds is a classic and fertile one.

Problem: give a metric description of a Riemannian manifold.

Background: A complete riemannian manifold is a length metric space (or geodesic, or intrinsic metric space) by Hopf-Rinow theorem. The problem asks for the recovery of the manifold structure from the distance function (associated to the length functional).

For 2-dim riemannian manifolds the problem has been solved by A. Wald [Begrundung einer koordinatenlosen Differentialgeometrie der Flachen, Erg. Math. Colloq. 7 (1936), 24-46] (“Begrundung” with umlaut u, “Flachen” with umlaut a, sorry for this).

In 1948 A.D. Alexandrov [Intrinsic geometry of convex surfaces, various editions] introduces its famous curvature (which uses comparison triangles)  and proves that, under mild smoothness conditions  on this curvature, one is capable to recover the differential structure and the metric of the 2-dim riemannian manifold. In 1982 Alexandrov proposes as a conjecture that a characterization of a riemannian manifold (of any dimension) is possible in terms of metric (sectional) curvatures (of the type introduced by Alexandrov) and weak smoothness assumptions formulated in metric way (as for example Holder smoothness). Many other results deserve to be mentioned (by Reshetnyak, for example).

2. Solution of the problem by Nikolaev. In 1998 I.G. Nikolaev [A metric characterization of riemannian spaces, Siberian Adv. Math. , 9 (1999), 1-58] solves the general problem of intrinsic characterization of $C^{m,\alpha}$ riemannian spaces:

every locally compact length metric space $M$, not linear at one of its points,  with $\alpha$ Holder continuous metric sectional curvature of the “generalized tangent bundle” $T^{m}(M)$ (for some $m=1,2,…$, which admits local geodesic extendability, is isometric to a $C^{m+2}$ smooth riemannian manifold..

Therefore:

• he defines a generalized tangent bundle in metric sense
• he defines a notion of sectional curvature
• he asks some metric smoothness of this curvature

and he gets the result.

3. Gleason metrics and Hilbert’s fifth problem. Let us compare this with the formulation of the solution of the Hilbert’s fifth problem by Terence Tao. THe problem is somehow similar, namely recover the differential structure of a Lie group from its algebraic structure. This time the “intrinsic” object is the group operation, not the distance, as previously.

Tao shows that the proof of the solution may be formulated in metric terms. Namely, he introduces a Gleason metric (definition 4 in the linked post), which will turn to be a left invariant riemannian metric on the (topological) group. I shall not insist on this, instead read the post of Tao and also, for the riemannian metric description, read this previous post by me.

Rules of lambda epsilon calculus

I updated the paper on lambda epsilon calculus.  See the link to the actual version (updated daily) or check the arxiv article, which will be updated as soon as a stable version will emerge.

Here are the rules of this calculus:

(beta *)   $(x \lambda A) \varepsilon B= (y \lambda (A[x:=B])) \varepsilon B$ for any fresh variable $y$,

(R1) (Reidemeister one) if $x \not \in FV(A)$ then $(x \lambda A) \varepsilon A = A$

(R2) (Reidemeister two) if $x \not \in FV(B)$ then $(x \lambda ( B \mu x)) \varepsilon A = B (\varepsilon \mu ) A$

(ext1) (extensionality one)  if  $x \not \in FV(A)$ then $x \lambda (A 1 x) = A$

(ext2) (extensionality two) if  $x \not \in FV(B)$ then $(x \lambda B) 1 A = B$

These are taken together with usual substitution and $\alpha$-conversion.

The relation between the operations from $\lambda \varepsilon$ calculus and emergent algebras is illustrated in the next figure.

Lambda epsilon calculus, an attempt for a language of “computing with space”

Just posted on the arxiv the paper Lambda calculus combined with emergent algebras.   Also, I updated my page on emergent algebras and computing with space, taking into consideration what I try to do in the mentioned paper. Here are some excerpts (from the mentioned page) about what I think now “computing with space” may be:

 Let’s look at some examples of spaces, like: the real world or a virtual world of a game, as seen by a fly or by a human, “abstract” mathematical spaces as manifolds, fractals, symmetric spaces, groups, linear spaces. To know what a space is, to define it mathematically, is less interesting than to know what one can do in such a space. I try to look at spaces from a kind of a “computational viewpoint”. A model of such a computation could be the following process: Alice and Bob share a class of primitives of spaces (like a common language which Alice can use in order to communicate to Bob what she sees when exploring the unknown space). Alice explores an unknown territory and sends to Bob the operational content of the exploration (i.e. maps of what she sees and how she moves, expresses in the common language ). Then Bob, who is placed in a familiar space, tries to make sense of the maps received from Alice. Usually, he can’t put together in the familiar the received information (for example because there is a limit of accuracy of maps of the unknown territory into the familiar one). Instead, Bob tries to simulate the operational content of Alice exploration by interpreting the messages from Alice (remember, expressed in a language of primitives of space) in his space. A language of space primitives could be (or contain as a part) the emergent algebras. Ideally, such a language of primitives should be described by: A – a class of gates (operations), which represent the “map-making” relation, with an abstract scale parameter attached, maybe also with supplementary operations (like lambda abstraction?), B – a class of variables (names) which represent generic points of a space, and a small class of terms (words) expressed in terms of variables and gates from (A) (resembling to combinators in lambda calculus?). These are the “generators” of the space and they have the emergent algebras property, namely that as the scale goes to zero, uniformly with respect to the variables, the output converges to a new operation. C – a class of rewriting rules saying that some simple assemblies of generators of space have equivalent function, or saying that relations between those simple assemblies converge to relations of the space as the scale goes to zero.

The article on lambda epsilon calculus is a first for me in this field, I would be grateful for any comments and suggestions of improvements.

Three problems and a disclaimer

In this post I want to summarize the list of problems I am currently thinking about. This is not a list of regular mathematical problems, see the disclaimer on style written at the end of the post.

Here is the list:

1. what is “computing with space“? There is something happening in the brain (of a human or of a fly) which is akin to a computation, but is not a logical computation: vision. I call this “computing with space”. In the head there are a bunch of neurons chirping one to another, that’s all. There is no euclidean geometry, there are no a priori coordinates (or other extensive properties), there are no problems to solve for them neurons, there is  no homunculus and no outer space, only a dynamical network of gates (neurons and their connections). I think that a part of an answer is the idea of emergent algebras (albeit there should be something more than this).  Mathematically, a closely related problem is this: Alice is exploring a unknown space and then sends to Bob enough information so that Bob could “simulate” the space in the lab. See this, or this, or this.

Application: give the smallest hint of a purely relational  model of vision  without using any a priori knowledge of the (euclidean or other) geometry of outer space or any  pre-defined charting of the visual system (don’t give names to neurons, don’t give them “tasks”, they are not engineers).

2. non-commutative Baker-Campbell-Hausdorff formula. From the solution of the Hilbert’s fifth problem we know that any locally compact topological group without small subgroups can be endowed with the structure of a “infinitesimally commutative” normed group with dilations. This is true because  one parameter sub-groups  and Gleason metrics are used to solve the problem.  The BCH formula solves then another problem: from the infinitesimal structure of a (Lie) group (that is the vector space structure of the tangent space at the identity and the maniflod structure of the Lie group) and from supplementary infinitesimal data (that is the Lie bracket), construct the group operation.

The problem of the non-commutative BCH is the following: suppose you are in a normed group with dilations. Then construct the group operation from the infinitesimal data (the conical group structure of the tangent space at identity and the dilation structure) and supplementary data (the halfbracket).

The classical BCH formula corresponds to the choice of the dilation structure coming from the manifold structure of the Lie group.

In the case of a Carnot group (or a conical group), the non-commutative BCH formula should be trivial (i.e. $x y = x \cdot y$, the equivalent of $xy = x+y$ in the case of a commutative Lie group, where by convention we neglect all “exp” and “log” in formulae).

3. give a notion of curvature which is meaningful for sub-riemannian spaces. I propose the pair curvdimension- curvature of a metric profile. There is a connection with problem 1: there is a link between the curvature of the metric profile and the “emergent Reidemeister 3 move” explained in section 6 of the computing with space paper. Indeed, at page 36 there is this figure. Yes, $R^{x}_{\epsilon \mu \lambda} (u,v) w$ is a curvature!

Disclaimer on style. I am not a problem solver, in the sense that I don’t usually like to find the solution of an already formulated problem. Instead, what I do like to do is to understand some phenomenon and prove something about it in the simplest way possible.  When thinking about a subject, I like to polish the partial understanding I have by renouncing to use any “impure” tools, that is any (mathematical) fact which is strange to the subject. I know that this is not the usual way of doing the job, but sometimes less is more.

Sub-riemannian geometry from intrinsic viewpoint, course notes

Here are the course notes prepared   for a course at CIMPA research school on sub-riemannian geometry (2012):

Sub-riemannian geometry from intrinsic viewpoint ( 27.02.2012) (14.06.2012)

I want to express my thanks for the invitation and also my excuses for not being abble to attend the school (due to very bad weather conditions in this part of Europe, I had to cancel my plane travel).

Approximate algebraic structures, emergent algebras

I updated and submitted for publication the paper “Emergent algebras“.

This is the first paper where emergent algebras appear. The subject is further developed in the paper “Braided spaces with dilations and sub-riemannian symmetric spaces“.

I strongly believe this is a very important notion, because it shows how  both the differential and algebraic realms  emerge naturally, from abstract nonsense. It is a “low tech” approach, meaning that I don’t use in the construction any “high tech” mathematical object, everything is growing from the grass.

One interesting fact, apart from the strange ideas of the paper (it is already verified that reading the paper algebraists will not understand easily the strength of the axiom concerning uniform convergence and analysts will not care enough about the occurrence of algebraic structure very much alike quandles and racks), is that an emergent algebra can also be seen as an approximate algebraic structure! But in a different sense than approximate groups.  The operations themselves are approximately associative, for example.

And my next question is: is this a really different notion of approximate algebraic structure than approximate groups? Or there is a way to see, for example, an approximate group (btw, why not an approximate symmetric space in the sense of Loos, whatever this could mean?) as an emergent algebra?

My hope is that the answer is YES.

UPDATE:   No, in fact there are reasons to think that there is a complementarity, there is a mathematical object standing over both, which may be called POSITIONAL SYSTEM, more soon, but see also this previous post of mine.

Here is the abstract of the paper:

“Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras.
Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example.
An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters.
Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property. “

On the difference of two Lipschitz functions defined on a Carnot group

Motivation for this post: the paper “Lipschitz and biLipschitz Maps on Carnot Groups” by William Meyerson. I don’t get it, even after several readings of the paper.

The proof of Fact 2.10 (page 10) starts by the statement that the difference of two Lipschitz functions is Lipschitz and the difference of two Pansu differentiable functions is differentiable.

Let us see: we have a Carnot group (which I shall assume is not commutative!) $G$ and two functions $f,g: U \subset G \rightarrow G$, where $U$ is an open set in $G$. (We may consider instead two Carnot groups $G$ and $H$ (both non commutative) and two functions $f,g: U \subset G \rightarrow H$.)

Denote by $h$ the difference of these functions: for any $x \in U$ $h(x) = f(x) (g(x))^{-1}$  (here the group operations  and inverses are denoted multiplicatively, thus if $G = \mathbb{R}^{n}$ then $h(x) = f(x) - g(x)$; but I shall suppose further that we work only in groups which are NOT commutative).

1.  Suppose $f$ and $g$ are Lipschitz with respect to the respective  CC left invariant distances (constructed from a choice of  euclidean norms on their respective left invariant distributions).   Is the function $h$ Lipschitz?

NO! Indeed, consider the Lipschitz functions $f(x) = x$, the identity function,  and $g(x) = u$ a constant function, with $u$ not in the center of $G$. Then $h$ is a right translation, notoriously NOT Lipschitz with respect to a CC left invariant distance.

2. Suppose instead that $f$ and $g$ are everywhere Pansu differentiable and let us compute the Pansu “finite difference”:

$(D_{\varepsilon} h )(x,u) = \delta_{\varepsilon^{-1}} ( h(x)^{-1} h(x \delta_{\varepsilon} u) )$

We get that $(D_{\varepsilon} h )(x,u)$ is the product w.r.t. the group operation of two terms: the first is the conjugation of the finite difference $(D_{\varepsilon} f )(x,u)$  by $\delta_{\varepsilon^{-1}} ( g(x) )$ and the second term is the finite difference   $(D_{\varepsilon} g^{-1} )(x,u)$.  (Here  $Inn(u)(v) = u v u^{-1}$ is the conjugation of $v$ by $u$ in the group $G$.)

Due to the non commutativity of the group operation, there should be some miracle in order for the finite difference of $h$ to converge, as $\varepsilon$ goes to zero.

We may take instead the sum of two differentiable functions, is it differentiable (in the sense of Pansu?). No, except in very particular situations,  because we can’t get rid of the conjugation, because the conjugation is not a Pansu differentiable function.

Non-euclidean analysis of dilation structures

Here are  the slides for the following talk:

Non-euclidean analysis of dilation structures

This is a general audience (of mathematicians) talk, a kind of appetizer for the subject.

Non-Euclidean analysis, a bit of recent history

Being an admirer of bold geometers who discovered that there is more to geometry than euclidean geometry, I believe that the same is true for analysis. In my first published paper “The topological substratum of the derivative” (here is a scan of this hard to find paper), back in 1993, I advanced the idea that there are as many “analyses” as the possible fields of dilations, but I was not aware about Pierre Pansu huge paper from 1989 “Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un” (sorry for the missing accents, I am still puzzled by the keyboard of the computer I am using to write this post), where he invents what is now called “Pansu calculus”, which is the analysis associated to a Carnot group.

The same idea is then explored in the papers “Sub-riemannian geometry and Lie groups, I“, “Tangent bundles to sub-riemannian groups“, “Sub-riemannian geometry and Lie groups II“. These papers have not been published (only put on arXiv), because at that moment I hoped that the www will change publishing quick (I still do believe this, but now I am just a bit wiser, or forced by bureaucracy to publish or perish), so one could communicate not only the very myopic technical, incremental result, but also the ideas behind, the powerful  meme.

During those years (2001-2005) I have been in Lausanne, trying to propagate the meme around, in Europe, as I said previously. There were mixed results, people were not taking this serious enough, according to my taste. Sergey Vodopyanov had ideas which were close to mine, except that he was trying to rely on what I call “euclidean analysis”, instead of intrinsic techniques, as witnessed by his outstanding results concerning detailed proofs in low-regularity sub-riemannian geometry. (I was against such results by principle, because what is $C^{1,1}$ but euclidean regularity? but the underlying ideas were very close indeed).

In a very naive way I tried to propagate the meme further, by asking for a visit at IHES, in 2004, when I had the pleasure to meet Pierre Pansu and Andre Bellaiche, then I dared to ask for another visit immediately and submitted the project

“Non-Euclidean Analysis” start-up

which I invite you to read. (The project was rejected, for good reasons, I was already there visiting and suddenly I was asking for another, much longer visit)

Then, from 2006 I went back to basics and proposed axioms for this, that is how dilation structures appeared (even if the name and a definition containing the most difficult axiom was already proposed in the previous series of papers on sub-riemannian geometry and Lie groups.  See my homepage for further details and papers (published this time).

I see now that, at least at the level of names of grant projects, the meme is starting to spread. Here is the “Sub-riemannian geometric analysis in Lie groups” GALA project and here is the more recent “Geometric measure theory in Non Euclidean spaces” GeMeThNES project.

Proof mining and approximate groups, announcement

The last weeks have been very busy for personal reasons. I shall come back to writing on this blog in short time.

With Laurentiu Leustean we won (a month ago, but the project has been submitted this Spring) the financing for our research project

“Proof mining in metric anaysis, geometric group theory and ergodic theory”

(project PN-II-ID-PCE-2011-3-0383). Laurentiu is a specialist in proof mining with applications to geodesic spaces and ergodic theory; I am interested in emergent algebras, particularly in dilation structures, so one of our aims (in this project) is to understand why nilpotent like structures appear in the famous Gromov theorem on groups of polynomial growth, as well in approximate groups, by using proof mining techniques for “finitizing” emergent algebras, roughly.

This program is very close to one of the programs of Terence Tao, who continues his outstanding research on approximate groups. The following post

Ultraproducts as a bridge between hard analysis and soft analysis

made me happy because it looks like confirming that our dreams (for the moment) have a correspondent in reality and probably ideas like this are floating in the air.

UPDATE 25.10: Today a new post of Tao announces the submission on arxiv of the paper by him, Emmanuel Breuillard and Ben Green, “The structure of approximate groups“. I look forward to study it, to see if they explain why nilpotent structures appear in the limit. My explanation, in a different context, “A characterization of sub-riemannian spaces…”, related also to work of Gromov, namely why in sub-riemannian geometry nilpotent groups appear as models of metric tangent spaces, is that this is a feature of an emergent algebra. See also previous posts, like Principles: randomness/structure or emergent from a common cause?.

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 3

This post continues the previous one “Gromov-Hausdorff distances and the Heisenberg group, PART 2“.

We have seen that small enough balls in physicist’ Heisenberg group $G$ are like balls in the mathematician’ Heisenberg group $H(1)$ and big balls in $G$ become more and more alike (asymptotically the same) as balls in the euclidean vector space $\mathbb{R}^{2}$.

What is causing this?

Could it be the choice of an euclidean norm on the generating set $D = \mathbb{R}^{2} \times \left\{ 1 \right\}$? I don’t think so, here is why. Let us take any (vector space) norm on $\mathbb{R}^{2}$, instead of an euclidean one. We may repeat all the construction and the final outcome would be: same for small balls, big balls become asymptotically alike to balls in $\mathbb{R}^{2}$ with the chosen norm. The algebraic structure of the limits in the infinitesimally small or infinitely big is the same.

Remember that the group norm is introduced only to estimate quantitatively how the set $D$ generates the group $G$, so the initial choice of the norm is a kind of a gauge.

Could it be then the algebraic structure (the group operation and choice of the generating set)? Yes, but there is much flexibility here.

Instead of $G = \mathbb{R}^{2} \times S^{1}$ with the given group operation, we may take any contact manifold structure over the set $G$ (technically we may take any symplectic structure over $\mathbb{R}^{2}$ and then contactify it (with the fiber $S^{1}$). Sounds familiar? Yes, indeed, this is a step in the recipe of geometric quantization. (If you really want to understand what is happening, then you should go and read Souriau).

Briefly said, put a norm on the kernel of the contact form and declare all directions in this kernel as horizontal, then repeat the construction of the sub-riemannian distance and metric profiles. What you get is this: small balls become asymptotically like balls in the mathematician’ Heisenberg group, big balls are alike balls in a normed vector space.

Therefore, it is not the algebraic structure per se which creates the phenomenon, but the “infinitesimal structure”. This will be treated in a later posting, but before this let me mention an amazing phenomenon.

We are again in the group $G$ and we want to make a map of the small (i.e. of a small enough ball in $G$) into the big (that is into a ball in the vector space $\mathbb{R}^{2}$, which is the asymptotically big model of balls from $G$). Our macroscopic lab is in the asymptotically big, while the phenomenon happens in the small.

A good map is a bi-lipschitz one (it respects the “gauges”, the group norm) from a ball in the vector space $\mathbb{R}^{2}$ to a ball in the Heisenberg group $H(1)$. Surprise: there is no such map! The reason is subtle, basically the same reason as the one which leads to the algebraic structure of the infinitesimally small or infinitely large balls.

However, there are plenty of bi-lipschitz maps from a curve in the ball from the lab (one dimensional submanifold of the symplectic $\mathbb{R}^{2}$, this are the lagrangian submanifolds in this case) to the small ball where the phenomenon happens. This is like: you can measure the position, or the momentum, but not both…

If there are not good bi-lipschitz maps, then there are surely quasi-isometric maps . Their accuracy is bounded by the Gromov-Hausdorff distance between big balls and small balls, as explained in this pedagogical Maps of metric spaces.

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 2

As the title shows, this post continues the previous one

Gromov-Hausdorff distances and the Heisenberg group, PART 1

The Heisenberg group $G$ is seen from the point of view of the generating set $D$. Quantitatively, the group norm “measures how” $D$ generates $G$. The group norm has the following properties:

• $\| X \| = 0$ if and only if $X = E = (0,1)$, the neutral element of $G$. In general $\| X\| \geq 0$ for any $X \in G$.
• $\| X \cdot Y \| \leq \|X\| + \|Y\|$, for any $X,Y \in G$ (that is a consequence of the fact that if we want to go from $E$ to $X \cdot Y$ by using horizontal increments, then we may go first from $E$ to $X$, then from $X$ to $X \cdot Y$, by using horizontal strings).
• $\| X^{-1} \| = \| X \|$ for any $X \in G$ (consequence of $X \in D$ implies $X^{-1} \in D$).

From (group) norms we obtain distances: by definition, the distance between $X$ and $Y$ is

$d(X,Y) = \| X^{-1} \cdot Y \|$

This is the sub-riemannian distance mentioned at the end of the previous post.

The definition of this distance does not say much about the properties of it. We may use a reasoning similar with the one in (finite dimensional) normed vector spaces in order to prove that any two group norms are equivalent. In our case, the result is the following:

there are strictly positive constants $a, c, C$ such that for any
$X \in G$ (which has the form $X = (x, e^{2\pi i z})$) with $\| X \| \leq a$ we have

$c ( x_{1}^{2} + x_{2}^{2} + \mid z \mid) \leq \|X\|^{2} \leq C ( x_{1}^{2} + x_{2}^{2} + \mid z \mid)$.

We may take $a = 1/3$, for example.

For “big” norms, we have another estimate, coming from the fact that the $S^{1}$ part of the semidirect product is compact, thus bounded:

there is a strictly positive constant $A$ such that for any $X \in G$ (which has the form $X = (x, e^{2\pi i z})$) we have

$\| x\| \leq \|X \| \leq \|x\| + A$

Let us look now at the ball $B(R) = \left\{ X \in G \mbox{ : } \|X\| \leq R \right\}$ endowed with the rescaled distance

$d_{R} (X,Y) = \frac{1}{R} d(X,Y)$

Denote by $Profile(R) = [B(R), d_{R}]$ the isometry class (the class of metric spaces isometric with … ) of $(B(R), d_{R})$. This is called a “metric profile”, see Introduction to metric spaces with dilations, section 2.3, for example.

The function which associates to $R > 0$ the $Profile(R)$ can be seen as a curve in the Gromov space of (isometry classes of) compact metric spaces, endowed with the Gromov-Hausdorff distance.

This curve parameterized with $R$ roams in this huge abstract space.
I want to see what happens when $R$ goes to zero or infinity. The interpretation is the following: when $R$ is small (or large, respectively), how the small (or large) balls look like?

Based on the previous estimates, we can answer this question.

When $R$ goes to infinity, the profile $Profile(R)$ becomes the one of the unit ball in $\mathbb{R}^{2}$ with the euclidean norm. Indeed, this is easy, because of the second estimate, which implies that for any $X = (R x, e^{2 \pi i z})$ and $Y = (R y, e^{2 \pi i u})$ which belong to $B(R)$, (thus $\|x\|, \|y\| \leq 1$) we have:

$d_{euclidean}(x, y) \leq d_{R}(X,Y) \leq d_{euclidean}(x, y) + \frac{A}{R}$.

Therefore, as $R$ goes to infinity, we get the isometry result.

On the other side, if $R$ is small enough (for example smaller or equal to $1/3$, then $Profile(R)$ becomes stationary!

Indeed, let me introduce a second Heisenberg group, baptized $H(1) = \mathbb{R}^{2} \times R$, with the group operation

$(x, z) \cdot (y, u) = (x+ y, z + u + \frac{1}{2}\omega(x,y))$

Remark that the function $(x, e^{2 \pi i z}) \mapsto (x,z)$ is a group morphism (in fact a local group isomorphism), for $z$ small enough! That means locally the groups $G$ and $H(1)$ are isomorphic. If you don’t know what a local group is then see the post Notes on local groups by Terence Tao.

By exactly the same procedure, we may put a group norm on $H(1)$.

OK, so small balls in $G$ are isometric with small balls in $H(1)$. What about the rescaling with $\frac{1}{R}$? Well, it turns out that the group $H(1)$ is selfsimilar, moreover, is a conical group (see for example section 6 from the paper Braided spaces with dilations… and check also the references, for the notion of conical group). Conical means that the group has a one parameter family of self-similarities: for any $R > 0$ the function

$\delta_{R} (x,z) = (R x, R^{2} z)$

is an auto-morphism of $H(1)$ and moreover:

$\| \delta_{R} (x,z) \| = R \| (x,z)\|$ for any $(x,z) \in H(1)$.

As a consequence, all balls in $H(1)$ look alike (i.e. the metric profile of the group $H(1)$ is stationary, a hallmark of null curvature…). More precisely, for any $R > 0$ and any $X,Y \in H(1)$, if we denote by $d$ the distance in $H(1)$ induced by the group norm, we have:

$d_{R}( \delta_{R} X, \delta_{R} Y) = d(X,Y)$.

Conclusion for this part: Small balls in $G$ look like balls in the Heisenberg group $H(1)$. Asymptotically, as $R$ goes to infinity, balls of radius $R$ in the group $G$ look more and more alike balls in the euclidean space $\mathbb{R}^{2}$ (notice that this space is self-similar as well, all balls are isometric, with distances properly rescaled).

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 1

This post continues the previous one “Quantum physics and the Gromov-Hausdorff distance“.

Let me take an example. We are in the following Heisenberg group (this is really a physicist Heisenberg group): the semidirect product $G = \mathbb{R}^{2} \times S^{1}$. Elements of the group have the form

$X = (x, e^{2\pi iz})$ with $x \in \mathbb{R}^{2}$ and $z \in \mathbb{R} / \mathbb{Z}$

(think that $z \in [0,1]$ with the identification $o = 1$).
The group operation is given by:

$X \cdot Y = (x,e^{2\pi i z}) \cdot (y, e^{2 \pi i u}) = (x+y, e^{2 \pi i (u+z+ \frac{1}{2} \omega(x,y))})$

where $\omega: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R}$ is the “symplectic form” (or area form)

$\omega(x,y) = x_{1} y_{2} - x_{2} y_{1}$

Remark 1. The pair $(\mathbb{R}^{2}, \omega)$ is a symplectic (linear) space. As we well know, the Hamiltonian mechanics lives in such spaces, therefore we may think about $(\mathbb{R}^{2}, \omega)$ as being a “classical”, or “large scale” (see further the justification, in PART 2) phase space of a mechanical system with one degree of freedom.

The group $G$ is generated by the subset $D = \mathbb{R}^{2} \times \left\{ 1 \right\}$, more precisely any element of $G$ can be expressed as a product of four elements of $D$ (in a non-unique way).

What is the geometry of $G$, seen as generated by $D$? In order to understand this, we may put an euclidean norm on $D$ (identified with $\mathbb{R}^{2}$):

$\| (x, 1) \| = \| x\|$, where $\|x\|^{2} = x_{1}^{2} + x_{2}^{2}$ for example.

Then we define “horizontal strings” and their “length”: a string $w = X_{1} ... X_{n}$ of elements of $G$ is horizontal if for any two successive elements of the string, say $X_{i}, X_{i+1}$ we have

$X_{i}^{-1} \cdot X_{i+1} \in D$, where $X^{-1}$ denotes the inverse of $X$ with respect to the group operation. Also, we have to ask that $X_{1} \in D$.

The length of the horizontal string $w = X_{1} ... X_{n}$ is defined as:

$l(w) = \|X_{1}\| + \| X_{1}^{-1} \cdot X_{2}\| + .... + \|X_{n-1}^{-1} \cdot X_{n}\|$. The source of the string $w$ is the neutral element $s(w) = E = (0,1)$ and the target of the string is $t(w) = X_{1}\cdot ... \cdot X_{n}$.

OK, then let us define the “group norm” of an element of $G$, which is an extension of the norm defined on $D$. A formula for this would be:

$\| X\| = \, inf \left\{ l(w) \mbox{ : } t(w) = X \right\}$.

Small technicality: it is not clear to me if this definition is really good as it is, but we may improve it by the following procedure coming from the definition of the Hausdorff measure. Let us introduce the “finesse” of a horizontal string, given by

$fin(w) = \max \left\{ \|X_{1}\| , \| X_{1}^{-1} \cdot X_{2}\| , ... , \|X_{n-1}^{-1} \cdot X_{n}\| \right\}$

and then define, for any $\varepsilon > 0$, the quantity:

$\| X\|_{\varepsilon} = \, inf \left\{ l(w) \mbox{ : } t(w) = X \mbox{ and } fin(w) < \varepsilon \right\}$.

The correct definition of the group norm is then

$\| X\| = \, sup \left\{\| X\|_{\varepsilon} \mbox{ : } \varepsilon > 0 \right\}$.

With words, that means: for a given “scale” $\varepsilon > 0$, take discrete paths from $E$ to $X$, made by “small” (norm smaller than $\varepsilon$) horizontal increments, and then take the infimum of the length of such curves. You get $\| X\|_{\varepsilon}$. Go with $\varepsilon$ to $o$ and get the norm $\| X\|_{\varepsilon}$.

Up to some normalization, the bigger is the norm of an element of $G$, the bigger is the infimal length of a horizontal curve which expresses it, therefore the group norm gives a quantitative estimate concerning how the group element is generated.

In disguise, this norm is nothing but a sub-riemannian distance!

Combinatorics versus geometric…

… is like using roman numerals versus using a positional numeral system, like the hindu-arabic numerals we all know very well. And there is evidence that our brain is NOT using combinatorial techniques, but geometric, see further.

What is this post about? Well, it is about the problem of using knowledge concerning topological groups in order to study discrete approximate groups, as Tao proposes in his new course, it is about discrete finitely generated groups with polynomial growth which, as Gromov taught us, when seen from far away they become nilpotent Lie groups, and so on. Only that there is a long way towards these subjects, so please bear me a little bit more.

This is part of a larger project to try to understand approximate groups, as well as normed groups with dilations, in a more geometric way. One point of interest is understanding the solution to the Hilbert’s fifth problem from a more general perspective, and this passes by NOT using combinatorial techniques from the start, even if they are one of the most beautiful mathematical gems which is the solution given by Gleason-Montgomery-Zippin to the problem.

What is combinatorial about this solution? Well, it reduces (in a brilliant way) the problem to counting, by using objects which are readily at hand in any topological group, namely the one-parameter subgroups. There is nothing wrong about this, only that, from this point of view, Gromov’s theorem on groups with polynomial growth appears as magical. Where is this nilpotent structure coming from?

As written in a previous post, Hilbert’s fifth problem without one-parameter subgroups, Gromov’ theorem is saying a profound geometrical thing about a finitely generated group with polynomial growth: that seen from far away this group is self-similar, that is a conical group, or a contractible group w.r.t. any of its dilations. That is all! the rest is just a Siebert’ result. This structure is deeply hidden in the proof and one of my purposes is to understand where it is coming from. A way of NOT understanding this is to use huge chunks of mathematical candy in order to make this structure appear by magic.

I cannot claim that I understand this, that I have a complete solution, but instead, for this post, I looked for an analogy and I think I have found one.

It is the one from the first lines of the post.

Indeed, what is wrong, by analogy, with the roman numeral system? Nothing, actually, we have generators, like I, V, X, L, C, D, M, and relations, like IIII = IV, and so on (yes, they are not generators and relations exactly like in a group sense). The problems appear when we want to do complex operations, like addition of large numbers. Then we have to be really clever and use very intelligent and profound combinatorial arguments in order to efficiently manage all the cancellations and whatnot coming from the massive use of relations. Relations are at very small scale, we have to bridge the way towards large scales, therefore we have to approximate the errors by counting in different ways and to be very clever about these ways.

Another solution for this, historically preferred, was to use a positional number system, which is more geometric, because it exploits a large scale property of natural numbers, which is that their collection is (approximately) self-similar. Indeed, take (as another kind of generators, again not in a group sense), a small set, like B={0, 1, 2, …, 8, 9} and count in base 10, which goes roughly like this: take a (big, preferably) natural number $X$ and do the following

– initialize $i = 1$,

– find the smallest natural power $a_{i}$ of 10 such that $10^{-a_{i}} X$ has a norm smaller than 10, then pick the element $k_{i}$ of $B$ which minimizes the distance to $10^{-a_{i}} X$,

– substract (from the right or the left, it does not matter here because addition of natural numbers is commutative) $10^{a_{i}} k_{i}$ from $X$, and rename the result by $X$,

-repeat until $X \in B$ and finish the algorithm by taking the last digit as $X$.

In the end (remember, I said “roughly”) represent $X$ as a string which codes the string of pairs $(a_{i}, k_{i})$.

The advantage of this representation of natural numbers is that we can do, with controlled precision, the addition of big numbers, approximately. Indeed, take two very big numbers $X, Y$ and take another number, like $10$. Then for any natural $n$ define

$(X+Y)$ approx(n) $= 10^{n} ( [X]_{n} + [Y]_{n})$

where $[X]_{n}$ is the number which is represented as the truncation of the string which represents $X$ up to the first $n$ letters, counting from left to right.
If $n$ is small compared with $X, Y$, then $(X+Y)$ approx(n) is close to the true $X+Y$, but the computational effort for calculating $(X+Y)$ approx(n) is much smaller than the one for calculating $X+Y$.

Once we have this large scale self-similarity, then we may exploit it by using the more geometrical positional numeral system instead of the roman numeral system, that is my analogy. Notice that in this (almost correct) algorithm $10^{a} X$ is not understood as $X+X+....+X$ $10^{a}$ times.

Let me now explain why the positional numeral system is more geometric, by giving a neuroscience argument, besides what I wrote in this previous post: “How not to get bored, by reading Gromov and Tao” (mind the comma!).

I reproduce from the wiki page on “Subitizing and counting

Subitizing, coined in 1949 by E.L. Kaufman et al.[1] refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective subitus (meaning “sudden”) and captures a feeling of immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range.[1] Number judgments for larger set-sizes were referred to either as counting or estimating, depending on the number of elements present within the display, and the time given to observers in which to respond (i.e., estimation occurs if insufficient time is available for observers to accurately count all the items present).

The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapid,[2] accurate[3] and confident.[4] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence.[1] In addition, response times rise in a dramatic fashion, with an extra 250–350 ms added for each additional item within the display beyond about four.[5]

This is a brain competence which is spatial (geometrical) in nature, as evidenced by Simultanagnosia:

Clinical evidence supporting the view that subitizing and counting may involve functionally and anatomically distinct brain areas comes from patients with simultanagnosia, one of the key components of Balint’s syndrome.[14] Patients with this disorder suffer from an inability to perceive visual scenes properly, being unable to localize objects in space, either by looking at the objects, pointing to them, or by verbally reporting their position.[14] Despite these dramatic symptoms, such patients are able to correctly recognize individual objects.[15] Crucially, people with simultanagnosia are unable to enumerate objects outside the subitizing range, either failing to count certain objects, or alternatively counting the same object several times.[16]

From the wiki description of simultanagnosia:

Simultanagnosia is a rare neurological disorder characterized by the inability of an individual to perceive more than a single object at a time. It is one of three major components of Bálint’s syndrome, an uncommon and incompletely understood variety of severe neuropsychological impairments involving space representation (visuospatial processing). The term “simultanagnosia” was first coined in 1924 by Wolpert to describe a condition where the affected individual could see individual details of a complex scene but failed to grasp the overall meaning of the image.[1]

I rest my case.

Baker-Campbell-Hausdorff polynomials and Menelaus theorem

This is a continuation of the previous post on the noncommutative BCH formula. For the “Menelaus theorem” part see this post.

Everything is related to “noncommutative techniques” for approximate groups, which hopefully will apply sometimes in the future to real combinatorial problems, like the Tao’ project presented here, and also to the problem of understanding curvature (in non-riemannian settings), see a hint here, and finally to the problem of higher order differential calculus in sub-riemannian geometry, see this for a comment on this blog.

Remark: as everything this days can be retrieved on the net, if you find in this blog something worthy to include in a published paper, then don’t be shy and mention this. I believe strongly in fair practices relating to this new age of scientific collaboration opened by the www, even if in the past too often ideas which I communicated freely were taken in published papers without attribution. Hey, I am happy to help! but unfortunately I have an ego too (not only an ergobrain, as any living creature).

For the moment we stay in a Lie group , with the convention to take the exponential equal to identity, i.e. to consider that the group operation can be written in terms of Lie brackets according to the BCH formula:

$x y = x + y + \frac{1}{2} [x,y] + \frac{1}{12}[x,[x,y]] - \frac{1}{12}[y,[y,x]]+...$

For any $\varepsilon \in (0,1]$ we define

$x \cdot_{\varepsilon} y = \varepsilon^{-1} ((\varepsilon x) (\varepsilon y))$

and we remark that $x \cdot_{\varepsilon} y \rightarrow x+y$ uniformly with respect to $x,y$ in a compact neighbourhood of the neutral element $e=0$. The BCH formula for the operation labeled with $\varepsilon$ is the following

$x \cdot_{\varepsilon} y = x + y + \frac{\varepsilon}{2} [x,y] + \frac{\varepsilon^{2}}{12}[x,[x,y]] - \frac{\varepsilon^{2}}{12}[y,[y,x]]+...$

$BCH^{0}_{\varepsilon} (x,y) = x \cdot_{\varepsilon} y$

and $BCH^{0}_{0}(x,y) = \lim_{\varepsilon \rightarrow 0} BCH^{0}_{\varepsilon}(x,y) = x + y$.

Define the “linearized dilation$\delta^{x}_{\varepsilon} y = x + \varepsilon (-x+y)$ (written like this on purpose, without using the commutativity of the “+” operation; due to limitations of my knowledge to use latex in this environment, I am shying away to put a bar over this dilation, to emphasize that it is different from the “group dilation”, equal to $x (\varepsilon(x^{-1}y))$).

Consider the family of $\beta > 0$ such that there is an uniform limit w.r.t. $x,y$ in compact set of the expression

$\delta_{\varepsilon^{-\beta}}^{BCH^{0}_{\varepsilon}(x,y)} BCH^{0}_{0}(x,y)$

and remark that this family has a maximum $\beta = 1$. Call this maximum $\alpha_{0}$ and define

$BCH^{1}_{\varepsilon}(x,y) = \delta_{\varepsilon^{-\alpha_{1}}}^{BCH^{0}_{\varepsilon}(x,y)} BCH^{0}_{0}(x,y)$

and $BCH^{1}_{0}(x,y) = \lim_{\varepsilon \rightarrow 0} BCH^{1}_{\varepsilon}(x,y)$.

Let us compute $BCH^{1}_{0}(x,y)$:

$BCH^{1}_{0}(x,y) = x + y + \frac{1}{2}[x,y]$

and also remark that

$BCH^{1}_{\varepsilon}(x,y) = x+y + \varepsilon^{-1} ( -(x+y) + (x \cdot_{\varepsilon} y))$.

We recognize in the right hand side an expression which is a relative of what I have called in the previous post an “approximate bracket”, relations (2) and (3). A better name for it is a halfbracket.

We may continue indefinitely this recipe. Namely for any natural number $i\geq 1$ we first define the maximal number $\alpha_{i}$ among all $\beta > 0$ with the property that the (uniform) limit exists

$\lim_{\varepsilon \rightarrow 0} \delta_{\varepsilon^{-\beta}}^{BCH^{i}_{\varepsilon}(x,y)} BCH^{i}_{0}(x,y)$

Generically we shall find $\alpha_{i} = 1$. We define then

$BCH^{i+1}_{\varepsilon}(x,y) = \delta_{\varepsilon^{-\alpha_{i}}}^{BCH^{i}_{\varepsilon}(x,y)} BCH^{i}_{0}(x,y)$

and $BCH^{i+1}_{0}(x,y) = \lim_{\varepsilon \rightarrow 0} BCH^{i+1}_{\varepsilon}(x,y)$.

It is time to use Menelaus theorem. Take a natural number $N > 0$. We may write (pretending we don’t know that all $\alpha_{i} = 1$, for $i = 0, ... N$):

$x \cdot_{\varepsilon} y = BCH^{0}_{\varepsilon}(x,y) = \delta^{BCH^{0}_{0}(x,y)}_{\varepsilon^{\alpha_{0}}} \delta^{BCH^{1}_{0}(x,y)}_{\varepsilon^{\alpha_{1}}} ... \delta^{BCH^{N}_{0}(x,y)}_{\varepsilon^{\alpha_{N}}} BCH^{N+1}_{\varepsilon}(x,y)$

Let us denote $\alpha_{0} + ... + \alpha_{N} = \gamma_{N}$ and introduce the BCH polynomial $PBCH^{N}(x,y)(\mu)$ (the variable of the polynomial is $\mu$), defined by: $PBCH^{N}(x,y)(\mu)$ is the unique element of the group with the property that for any other element $z$ (close enough to the neutral element) we have

$\delta^{BCH^{0}_{0}(x,y)}_{\mu^{\alpha_{0}}} \delta^{BCH^{1}_{0}(x,y)}_{\mu^{\alpha_{1}}} ... \delta^{BCH^{N}_{0}(x,y)}_{\mu^{\alpha_{N}}} z = \delta^{PBCH^{N}(x,y)(\mu)}_{\mu^{\gamma_{N}}} z$

Such an element exists and it is unique due to (Artin’ version of the) Menelaus theorem.

Remark that $PBCH^{N}(x,y)(\mu)$ is not a true polynomial in $\mu$, but it is a rational function of $\mu$ which is a polynomial up to terms of order $\mu^{\gamma_{N}}$. A straightforward computation shows that the BCH polynomial (up to terms of the mentioned order) is a truncation of the BCH formula up to terms containing $N-1$ brackets, when we take $\mu =1$.

It looks contorted, but written this way it works verbatim for normed groups with dilations! There are several things which are different in detail. These are:

1. the coefficients $\alpha_{i}$ are not equal to $1$, in general. Moreover, I can prove that the $\alpha_{i}$ exist (as a maximum of numbers $\beta$ such that …) for a sub-riemannian Lie group, that is for a Lie group endowed with a left-invariant dilation structure, by using the classical BCH formula, but I don’t think that one can prove the existence of these numbers for a general group with dilations! Remark that the numbers $\alpha_{i}$ are defined in a similar way as Hausdorff dimension is!

2. one has to define noncommutative polynomials, i.e. polynomials in the frame of Carnot groups (at least). This can be done, it has been sketched in a previous paper of mine, Tangent bundles to sub-riemannian groups, section 6.

UPDATE: (30.10.2011) See the post of Tao

Associativity of the Baker-Campbell-Hausdorff formula

where a (trained) eye may see the appearance of several ingredients, in the particular commutative case, of the mechanism of definition of the BCH formula.

The associativity is rephrased, in a well known way,  in proposition 2 as a commutativity of say left and  right actions. From there signs of commutativity (unconsciously assumed) appear:  the obvious first are the “radial  homogeneity  identities”, but already at this stage a lot of familiar  machinery is put in place and the following is more and more heavy of  the same. I can only wonder:  is this  all necessary? My guess is: not. Because for starters, as explained here and in previous posts, Lie algebras are of a commutative blend, like the BCH formula. And (local, well known from the beginning) groups are not.

Principles: randomness/structure or emergent from a common cause?

I think both.

Finally Terence Tao presented a sketch of his project relating Hilbert’ fifth problem and approximate groups. For me the most interesting part is his Theorem 12 and its relation with Gromov’ theorem on finitely generated groups with polynomial growth.

A bit dissapointing (for me) is that he seems to choose to rely on “commutative” techniques and, not surprising, he is bound to get results valid only in riemannian geometry (or spaces of Alexander type) AND also to regard the apparition of nilpotent structures as qualitatively different from smooth structures.

For clarity reasons, I copy-paste his two “broad principles”

The above three families of results exemplify two broad principles (part of what I like to call “the dichotomy between structure and randomness“):

• (Rigidity) If a group-like object exhibits a weak amount of regularity, then it (or a large portion thereof) often automatically exhibits a strong amount of regularity as well;
• (Structure) This strong regularity manifests itself either as Lie type structure (in continuous settings) or nilpotent type structure (in discrete settings). (In some cases, “nilpotent” should be replaced by sister properties such as “abelian“, “solvable“, or “polycyclic“.)

Let me contrast with my

Principle of common cause: an uniformly continuous algebraic structure has a smooth structure because both structures can be constructed from an underlying emergent algebra (introduced here).

(from this previous post).

While his “dichotomy between structure and randomness“ principle is a very profound and mysterious one, the second part (Structure) is only an illusion created by psychological choices (I guess). Indeed, both (Lie) smooth structure and nilpotence are just “noncommutative (local) linearity”, as explained previously. Both smooth structure and “conical algebraic” structure (nilpotent in particular) stem from the same underlying dilation structure. What is most mysterious, in my opinion, be it in the “exact” or “approximate” group structure, is HOW a (variant of) dilation structure appears from non-randomness (presumably as a manifestation of Tao randomness/structure principle), i.e. HOW just a little bit of approximate self-similarity bootstraps to a dilation structure, or emergent algebra, which then leads to various flavors of “smoothness” and “nilpotence”.

Menelaus theorem by way of Reidemeister move 3

(Half of) Menelaus theorem is equivalent with a theorem by Emil Artin, from his excellent book Geometric Algebra, Interscience Publishers (1957), saying that the inverse semigroup generated by dilations (in an affine space) is composed by dilations and translations. More specifically, if $\varepsilon, \mu > 0$ are such that $\varepsilon \mu < 1$ then the composition of two dilations, one of coefficient $\varepsilon$ and the other of coefficient $\mu$, is a dilation of coefficient $\varepsilon \mu$.

Artin contributed also to braid theory, so it may be a nice idea to give a proof of Artin interpretation of Menelaus theorem by using Reidemeister moves.

This post is related to previous ones, especially these three:

Noncommutative Baker-Campbell-Hausdorff formula

A difference which makes four differences, in two ways

Rigidity of algebraic structure: principle of common cause

which I shall use as references for what a normed group with dilations, conical group and associated decorations of tangle diagrams are.

Let’s start! I use a representation of dilations as decorations of an oriented tangle diagram.

For any $\varepsilon > 0$, dilations of coefficient $\varepsilon$ and $\varepsilon^{-1}$ provide two operations which give to the space (say $X$) the structure of an idempotent right quasigroup, which is equivalent to saying that decorations of the tangle diagrams by these rules are stable to the Reidemeister moves of type I and II.

A particular example of a space with dilations is a normed group with dilations, where the dilations are left-invariant.

If the decorations that we make are also stable with respect to the Reidemeister move 3, then it can be proved that definitely the space with dilations which I use has to be a conical group! What is a conical group? It is a non-commutative vector space, in particular it could be a real vector space or the Heisenberg group, or a Carnot group and so on. Read the previous posts about this.

Graphically, the Reidemeister move 3 is this sliding movement:

of $CC'$ under the crossing $AA'-BB'$ (remark also how the decorations of crossings $\varepsilon$ and $\mu$ switch places).
Further on I shall suppose that we use for decorations a conical group, with distance function denoted by $d$. Think about a real vector space with distance given by an euclidean norm, but don’t forget that in fact we don’t need to be so restrictive.

Take now two strings and twist them one around the other, an infinity of times, then pass a third string under the first two, then decorate everything as in the following figure

We can slide twice the red string (the one which is under) by using the Reidemeister move 3. The decorations do not change. If you want to see what is the expression of $z'$ as a function of $x,y$ then we easily write that

$z' = \delta^{x}_{\varepsilon} \delta^{y}_{\mu} z = \delta^{x_{1}}_{\varepsilon} \delta^{y_{1}}_{\mu} z$

where $x_{1}. y_{1}$ are obtained from $x,y$ according to the rules of decorations.

We may repeat $n$ times the double slide movement and we get that

$z' = \delta^{x_{n}}_{\varepsilon} \delta^{y_{n}}_{\mu} z$

If we prove that the sequences $x_{n}, y_{n}$ converge both to some point $w$, then by passing to the limit in the previous equality we would get that

$z' = \delta^{w}_{\varepsilon} \delta^{w}_{\mu} z = \delta^{w}_{\varepsilon \mu} z$

which is the conclusion of Artin’s result! Otherwise said, if we slide the red string under all the twisted pair of strings, then the outcome is the same as passing under only one string, decorated with $w$, with the crossing decorated with $\varepsilon \mu$.

The only thing left is to prove that indeed the sequences converge. But this is easy: we prove that the recurrence relation between $x_{n+1}, y_{n+1}$ and $x_{n},y_{n}$ is a contraction. See the figure:

Well, that is all.

UPDATE: I was in fact motivated to draw the figures and explain all this after seeing this very nice post of Tao, where an elementary proof of a famous result is given, by using “elementary” graphical means.

How not to get bored, by reading Gromov and Tao

This is a continuation of the previous post Gromov’s ergobrain, triggered by the update of Gromov paper on July 27. It is also related to the series of posts by Tao on the Hilbert’s fifth problem.

To put you in the right frame of mind, both Gromov and Tao set the stage for upcoming, hopefully extremely interesting (or “beautiful” on Gromov’s scale: interesting, amusing, amazing, funny and beautiful) developments of their work on “ergosystems” and “approximate groups” respectively.

What can be the link between those? In my opinion, both works refer to the unexplored ground between discrete (with not so many elements) and continuous (or going to the limit with the number of elements of a discrete world).

Indeed, along with my excuses for simplifying too much a very rich text, let me start with the example of the bug on a leaf, sections 2.12, 2.13 in Gromov’s paper). I understand that the bug, as any other “ergosystem” (like one’s brain) would get bored to behave like a finite state automaton crawling on a “co-labeled graph” (in particular on a Cayley graph of a discretely generated group). The implication seems to be that an ergosystem has a different behaviour.

I hardly refrain to copy-paste the whole page 96 of Gromov’s paper, please use the link and read it instead, especially the part related to downsides of Turing modeling (it is not geometrical, in few words). I shall just paste here the end:

The two ergo-lessons one may draw from Turing models are mutually contradictory.
1. A repeated application of a simple operation(s) may lead to something unexpectedly complicated and interesting.
2. If you do not resent the command “repete” and/or are not getting bored by doing the same thing over and over again, you will spend your life in a “Turing loop” of an endless walk in a circular tunnel.

That is because the “stop-function” associated to a class of Turing machines

may grow faster than anything you can imagine, faster than anything expressible by any conceivable formula – the exponential and double exponential functions that appeared so big to you seem as tiny specks of dust compared to this monstrous stop-function. (page 95)

Have I said “Cayley graph”? This brings me to discrete groups and to the work of Tao (and Ben Green and many others). According to Tao, there is something to be learned from the solution of the Hilbert’s fifth problem, in the benefit of understanding approximate groups. (I am looking forward to see this!) There are some things that I understood from the posts of Tao, especially that a central concept is a Gleason metric and its relations with group commutators. In previous posts (last is this) I argue that Gleason metrics are very unlike sub-riemannian distances. It has been unsaid, but obvious to specialists, that sub-riemannian metrics are just like distances on Cayley graphs, so as a consequence Gleason metrics are only a commutative “shadow” of what happens in a Cayley graph when looked from afar. Moreover, in this post concerning the problem of a non-commutative Baker-Campbell-Hausdorff formula it is said that (in the more general world of groups with dilations, relevant soon in this post) the link between the Lie bracket and group commutators is shallow and due to the commutativity of the group operation in the tangent space.

So let me explain, by using Gromov’s idea of boredom, how not to get bored in a Cayley graph. Remember that I quoted a paragraph (from Gromov paper, previous version), stating that an ergosystem “would be bored to death” to add large numbers? Equivalently, an ergosystem would be bored to add (by using the group operation) elements of the group expressed as very long words with letters representing the generators of the group. Just by using “finite state automata” type of reasoning with the relations between generators (expressed by commutators and finitary versions of Gleason like metrics) an ergosystem would get easily bored. What else can be done?

Suppose that we crawl in the Cayley graph of a group with polynomial growth, therefore we know (by a famous result of Gromov) that seen from afar the group is a nilpotent one, more precisely a group with the algebraic structure completely specified by its dilations. Take one such dilation, of coefficient $10^{-30}$ say, and (by an yet unknown “finitization” procedure) associate to it a “discrete shadow”, that is an “approximate dilation” acting on the discrete group itself. As this is a genuinely non-commutative object, probably the algorithm for defining it (by using relations between growth and commutators) would be very much resource consuming. But suppose we just have it, inferred from “looking at the forrest” as an ergosystem.

What a great object would that be. Indeed, instead of getting bored by adding two group elements, the first expressed as product of 200034156998123039234534530081 generators, the second expressed as a product of 311340006349200600380943586878 generators, we shall first reduce the elements (apply the dilation of coefficient $10^{-30}$) to a couple of elements, first expressed as a product of 2 generators, second expressed as a product of 3 generators, then we do the addition $2+3 = 5$ (and use the relations between generators), then we use the inverse dilation (which is a dilation of coefficient $10^{30}$) to obtain the “approximate sum” of the two elements!

In practice, we probably have a dilation of coefficient $1/2$ which could simplify the computation of products of group elements of length $2^{4}$ at most, for example.

But it looks like a solution to the problem of not getting bored, at least to me.

Braitenberg vehicles, enchanted looms and winnowing-fans

Braitenberg vehicles were introduced in the wonderful book (here is an excerpt which contains enough information for understanding this post):

Vehicles: Experiments in Synthetic Psychology [update: link no longer available]

In the introduction of the book we find the following:

At times, though, in the back of my mind, while I was counting fibers in the visual ganglia of the fly or synapses in the cerebral cortex of the mouse, I felt knots untie,  distinctions dissolve, difficulties disappear, difficulties I had experienced much earlier when I still held my first naive philosophical approach to the problem of the mind.

This is not the first appearance of knots (and related weaving craft) as a metaphor for things related to the brain. A famous paragraph, by Charles Scott Sherrington compares the brain waking from sleep with an enchanted loom

The great topmost sheet of the mass, that where hardly a light had twinkled or moved, becomes now a sparkling field of rhythmic flashing points with trains of traveling sparks hurrying hither and thither. The brain is waking and with it the mind is returning. It is as if the Milky Way entered upon some cosmic dance. Swiftly the head mass becomes an enchanted loom where millions of flashing shuttles weave a dissolving pattern, always a meaningful pattern though never an abiding one; a shifting harmony of subpatterns.

Compare with the following passage (Timaeus 52d and following) from Plato:

…the nurse of generation [i.e. space, chora] …  presented a strange variety of appearances; and being full of powers which were neither similar nor equally balanced, was never in any part in a state of equipoise, but swaying unevenly hither and thither, was shaken by them, and by its motion again shook them; and the elements when moved were separated and carried continually, some one way, some another; as, when grain is shaken and winnowed by fans and other instruments used in the threshing of corn, the close and heavy particles are borne away and settle in one direction, and the loose and light particles in another.

The winnowing-fan (liknon) is important in the Greek mythology, it means also cradle and Plato uses this term with both meanings.

For a mathematician at least, winnowing and weaving are both metaphors of computing with braids: the fundamental group of the configuration space of the grains is the braid group and moreover the grains (trajectories) are the weft, the winnowing-fan is the warp of a loom.

All part of the reason of proposing a tangle formalism for chora and computing with space.

Back to Braitenberg vehicles. Vehicles 2,3,4 and arguably 5 are doing computations with space, not logical computations, by using sensors, motors and connections (that is map-making operations). I cite from the end of Vehicle 3 section:

But, you will say, this is ridiculous: knowledge implies a flow of information from the environment into a living being ar at least into something like a living being. There was no such transmission of information here. We were just playing with sensors, motors and connections: the properties that happened to emerge may look like knowledge but really are not. We should be careful with such words. […]

Meanwhile I invite you to consider the enormous wealth of different properties that we may give Vehicle 3c by choosing various sensors and various combinations of crossed and uncrossed, excitatory and inhibitory, connections.

Two papers on arXiv

I put on arxiv two papers

The paper Computing with space contains too may ideas, is too dense, therefore much of it will not be read, as I was warned repeatedly. This is the reason to do again what I did with Introduction to metric spaces with dilations, which is a slightly edited part of the paper A characterization of sub-riemannian spaces as length dilation structures. Apparently the part (Introduction to ..), the small detail,  is much more read  than the whole (A characterization…).

Concerning the second paper “Normed groupoids…”, it is an improvement of the older paper. Why did I not updated the older paper? Because I need help, I just don’t understand where this is going (and why such direction of research was not explored before).

Escape property of the Gleason metric and sub-riemannian distances again

The last post of Tao from his series of posts on the Hilbert’s fifth problem contains interesting results which can be used for understanding the differences between Gleason distances and sub-riemannian distances or, more general, norms on groups with dilations.

For normed groups with dilations see my previous post (where links to articles are also provided). Check my homepage for more details (finally I am online again).

There is also another post of mine on the Gleason metric (distance) and the CC (or sub-riemannian) distance, where I explain why the commutator estimate (definition 3, relation (2) from the last post of Tao) forces “commutativity”, in the sense that a sub-riemannian left invariant distance on a Lie group which has the commutator estimate must be a riemannian distance.

What about the escape property (Definition 3, relation (1) from the post of Tao)?

From his Proposition 10 we see that the escape property implies the commutator estimate, therefore a sub-riemannian left invariant distance with the escape property must be riemannian.

An explanation of this phenomenon can be deduced by using the notion of “coherent projection”, section 9 of the paper

A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111

in the very particular case of sub-riemannian Lie groups (or for that matter normed groups with dilations).

Suppose we have a normed group with dilations $(G, \delta)$ which has another left invariant dilation structure on it (in the paper this is denoted by a “$\delta$ bar”, here I shall use the notation $\alpha$ for this supplementary dilation structure).

There is one such a dilation structure available for any Lie group (notice that I am not trying to give a proof of the H5 problem), namely for any $\varepsilon > 0$ (but not too big)

$\alpha_{\varepsilon} g = \exp ( \varepsilon \log (g))$

(maybe interesting: which famous lemma is equivalent with the fact that $(G,\alpha)$ is a group with dilations?)
Take $\delta$ to be a dilation structure coming from a left-invariant distribution on the group . Then $\delta$ commutes with $\alpha$ and moreover

(*) $\lim_{\varepsilon \rightarrow 0} \alpha_{\varepsilon}^{-1} \delta_{\varepsilon} x = Q(x)$

where $Q$ is a projection: $Q(Q(x)) = x$ for any $x \in G$.

It is straightforward to check that (the left-translation of) $Q$ (over the whole group) is a coherent projection, more precisely it is the projection on the distribution!

Exercise: denote by $\varepsilon = 1/n$ and use (*) to prove that the escape property of Tao implies that $Q$ is (locally) injective. This implies in turn that $Q = id$, therefore the distribution is the tangent bundle, therefore the distance is riemannian!

UPDATE:    See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for understanding in detail the role of the escape property in the proof of the Hilbert 5th problem.

Pros and cons of higher order Pansu derivatives

This interesting question from mathoverflow

Higher order Pansu derivative

is asked by nil (no website, no location). I shall try to explain the pros and cons of higher order derivatives in Carnot groups. As for a real answer to nil’s question, I could tell him but then …

For “Pansu derivative” see the paper: (mentioned in this previous post)

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, The Annals of Mathematics Second Series, Vol. 129, No. 1 (Jan., 1989), pp. 1-60

Such derivatives can be done in any metric space with dilations, or in any normed group with dilations in particular (see definition in this previous post).

Pros/cons: It would be interesting to have a higher order differential calculus with Pansu derivatives, for all the reasons which make higher derivatives interesting in more familiar situations. Three examples come to my mind: convexity, higher order differential operators and curvature.

1. Convexity pro: the positivity of the hessian of a function implies convexity. In the world of Carnot groups the most natural definition of convexity (at least that is what I think) is the following: a function $f: N \rightarrow \mathbb{R}$, defined on a Carnot group $N$ with (homogeneous) dilations $\displaystyle \delta_{\varepsilon}$, is convex if for any $x,y \in N$ and for any $\varepsilon \in [0,1]$ we have

$f( x \delta_{\varepsilon}(x^{-1} y)) \leq f(x) + \varepsilon (-f(x) + f(y))$.

There are conditions in terms of higher order horizontal derivatives (if the function is derivable in the classical sense) which are sufficient for the function to be convex (in the mentioned sense). Note that the positivity of the horizontal hessian is not enough! It would be nice to have a more intrinsic differential condition, which does not use classical horizontal derivatives. Con: as in classical analysis, we can do well without second order derivatives when we study convexity. In fact convex analysis is so funny because we can do it without the need of differentiability.

2. Differential operators Pro: Speaking about higher order horizontal derivatives, notice that the horizontal laplacian is not expressed in an intrinsic manner (i.e. as a combinaion of higher order Pansu derivatives). It would be interesting to have such a representation for the horizontal laplacian, at least for not having to use “coordinates” (well, these are families of horizontal vector fields which span the distribution) in order to be able to define the operator. Con: nevertheless the horizontal hessian can be defined intrinsically in a weak sense, using only the sub-riemannian distance (and the energy functional associated to it, as in the classical case). Sobolev spaces and others are a flourishing field of research, without the need to appeal to higher order Pansu derivatives. (pro: this regards the existence of solutions in a weak sense, but to be honest, what about the regularity business?)

3. Curvature Pro: What is the curvature of a level set of a function defined on a Carnot group? Clearly higher order derivatives are needed here. Con: level set are not even rectifiable in the Carnot world!

Besides all this, there is a general:

Con: There are not many functions, from a Carnot group to itself, which are Pansu derivable everywhere, with continuous derivative. Indeed, for most Carnot groups (excepting the Heisenberg type and the jet type) only left translations are “smooth” in this sense. So even if we could define higher order derivatives, there is not much room to apply them.

However, I think that it is possible to define derivatives of Pansu type such that always there are lots of functions derivable in this sense and moreover it is possible to introduce higher order derivatives of Pansu type (i.e. which can be expressed with dilations).

UPDATE:  This should be read in conjunction with this post. Please look at Lemma 11   from the   last post of Tao    and also at the notations made previously in that post.  Now, relation (4) contains an estimate of a kind of discretization of a second order derivative. Based on Lemma 11 and on what I explained in the linked post, the relation (4) cannot hold in the sub-riemannian world, that is there is surely no bump  function $\phi$ such that $d_{\phi}$ is equivalent with a sub-riemannian distance (unless the metric is riemannian). In conclusion, there are no “interesting” nontrivial $C^{1,1}$ bump functions (say quadratic-like, see in the post of Tao how he constructs his bump function by using the distance).

There must be something going wrong with the “Taylor expansion” from the end of the proof of Lemma 11,  if instead of a norm with respect to a bump function we put a sub-riemannian distance. Presumably instead of “$n$”  and  “$n^{2}$” we have to put something else, like   “$n^{a}$”    and  “$n^{b}$” respectively, with coefficients  $a, b/2 <1$ and also functions of (a kind of  degree,  say) of $g$. Well, the coefficient $b$ will be very interesting, because related to some notion of curvature to be discovered.

Noncommutative Baker-Campbell-Hausdorff formula: the problem

I come back to a problem alluded in a previous post, where the proof of the Baker-Campbell-Hausdorff formula from this post by Tao is characterized as “commutative”, because of the “radial homogeneity” condition in his Theorem 1 , which forces commutativity.

Now I am going to try to explain this, as well as what the problem of a “noncommutative” BCH formula would be.

Take a Lie group $G$ and identify a neighbourhood of its neutral element with a neighbourhood of the $0$ element of its Lie algebra. This is standard for Carnot groups (connected, simply connected nilpotent groups which admit a one parameter family of contracting automorphisms), where the exponential is bijective, so the identification is global. The advantage of this identification is that we get rid of log’s and exp’s in formulae.

For every $s > 0$ define a deformation of the group operation (which is denoted multiplicatively), by the formula

(1)                $s(x *_{s} y) = (sx) (sy)$

Then we have $x *_{s} y \rightarrow x+y$ as $s \rightarrow 0$.

Denote by $[x,y]$ the Lie bracket of the (Lie algebra of the) group $G$ with initial operation and likewise denote by $[x,y]_{s}$ the Lie bracket of the operation $*_{s}$.

The relation between these brackets is: $[x,y]_{s} = s [x,y]$.

From the Baker-Campbell-Hausdorff formula we get:

$-x + (x *_{s} y) - y = \frac{s}{2} [x,y] + o(s)$,

(for reasons which will be clear later, I am not using the commutativity of addition), therefore

(2)         $\frac{1}{s} ( -x + (x *_{s} y) - y ) \rightarrow \frac{1}{2} [x,y]$       as        $s \rightarrow 0$.

Remark that (2) looks like a valid definition of the Lie bracket which is not related to the group commutator. Moreover, it is a formula where we differentiate only once, so to say. In the usual derivation of the Lie bracket from the group commutator we have to differentiate twice!

Let us now pass to a slightly different context: suppose $G$ is a normed group with dilations (the norm is for simplicity, we can do without; in the case of “usual” Lie groups, taking a norm corresponds to taking a left invariant Riemannian distance on the group).

$G$ is a normed group with dilations if

• it is a normed group, that is there is a norm function defined on $G$ with values in $[0,+\infty)$, denoted by $\|x\|$, such that

$\|x\| = 0$ iff $x = e$ (the neutral element)

$\| x y \| \leq \|x\| + \|y\|$

$\| x^{-1} \| = \| x \|$

– “balls” $\left\{ x \mid \|x\| \leq r \right\}$ are compact in the topology induced by the distance $d(x,y) = \|x^{-1} y\|$,

• and a “multiplication by positive scalars” $(s,x) \in (0,\infty) \times G \mapsto sx \in G$ with the properties:

$s(px) = (sp)x$ , $1x = x$ and $sx \rightarrow e$ as $s \rightarrow 0$; also $s(x^{-1}) = (sx)^{-1}$,

– define $x *_{s} y$ as previously, by the formula (1) (only this time use the multiplication by positive scalars). Then

$x *_{s} y \rightarrow x \cdot y$      as      $s \rightarrow 0$

uniformly with respect to $x, y$ in an arbitrarry closed ball.

$\frac{1}{s} \| sx \| \rightarrow \|x \|_{0}$, uniformly with respect to $x$ in a closed ball, and moreover $\|x\|_{0} = 0$ implies $x = e$.

1. In truth, everything is defined in a neighbourhood of the neutral element, also $G$ has only to be a local group.

2. the operation $x \cdot y$ is a (local) group operation and the function $\|x\|_{0}$ is a norm for this operation, which is also “homogeneous”, in the sense

$\|sx\|_{0} = s \|x\|_{0}$.

Also we have the distributivity property $s(x \cdot y) = (sx) \cdot (sy)$, but generally the dot operation is not commutative.

3. A Lie group with a left invariant Riemannian distance $d$ and with the usual multiplication by scalars (after making the identification of a neighbourhood of the neutral element with a neighbourhood in the Lie algebra) is an example of a normed group with dilations, with the norm $\|x\| = d(e,x)$.

4. Any Carnot group can be endowed with a structure of a group with dilations, by defining the multiplication by positive scalars with the help of its intrinsic dilations. Indeed, take for example a Heisenberg group $G = \mathbb{R}^{3}$ with the operation

$(x_{1}, x_{2}, x_{3}) (y_{1}, y_{2}, y_{3}) = (x_{1} + y_{1}, x_{2} + y_{2}, x_{3} + y_{3} + \frac{1}{2} (x_{1}y_{2} - x_{2} y_{1}))$

multiplication by positive scalars

$s (x_{1},x_{2},x_{3}) = (sx_{1}, sx_{2}, s^{2}x_{3})$

and norm given by

$\| (x_{1}, x_{2}, x_{3}) \|^{2} = (x_{1})^{2} + (x_{2})^{2} + \mid x_{3} \mid$

Then we have $X \cdot Y = XY$, for any $X,Y \in G$ and $\| X\|_{0} = \|X\|$ for any $X \in G$.

Carnot groups are therefore just a noncommutative generalization of vector spaces, with the addition operation $+$ replaced by a noncommutative operation!

5. There are many groups with dilations which are not Carnot groups. For example endow any Lie group with a left invariant sub-riemannian structure and hop, this gives a norm group with dilations structure.

In such a group with dilations the “radial homogeneity” condition of Tao implies that the operation $x \cdot y$ is commutative! (see the references given in this previous post). Indeed, this radial homogeneity is equivalent with the following assertion: for any $s \in (0,1)$ and any $x, y \in G$

$x s( x^{-1} ) = (1-s)x$

which is called elsewhere “barycentric condition”. This condition is false in any noncommutative Carnot group! What it is true is the following: let, in a Carnot group, $x$ be any solution of the equation

$x s( x^{-1} ) = y$

for given $y \in G$ and $s \in (0,1)$. Then

$x = \sum_{k=0}^{\infty} (s^{k}) y$ ,

(so the solution is unique) where the sum is taken with respect to the group operation (noncommutative series).

Problem of the noncommutative BCH formula: In a normed group with dilations, express the group operation $xy$ as a noncommutative series, by using instead of “$+$” the operation “$\cdot$” and by using a definition of the “noncommutative Lie bracket” in the same spirit as (2), that is something related to the asymptotic behaviour of the “approximate bracket”

(3)         $[x,y]_{s} = (s^{-1}) ( x^{-1} \cdot (x *_{s} y) \cdot y^{-1} )$.

Notice that there is NO CHANCE to have a limit like the one in (2), so the problem seems hard also from this point of view.

Also notice that if $G$ is a Carnot group then

$[x,y]_{s} = e$ (that is like it is equal to $o$, remember)

which is normal, if we think about $G$ as being a kind of noncommutative vector space, even of $G$ may be not commutative.

So this noncommutative Lie bracket is not about commutators!

Topological substratum of the derivative

The topological substratum of the derivative (I), Math. Reports (Stud. Cerc. Mat.) 45, 6,       (1993), 453-465

which is no longer visible now. But maybe it deserves a post here, because is my oldest attempt to understand differential calculus as an abstract matter and to look to new forms of it.

To me it became clear that differential calculus admits variants, in the same spirit as euclidean geometry admitting non-euclidean variants. At that moment I had no really intersting examples of such a “non-euclidean” differential calculus, so I switched to other research subjects. Nobody pointed to me the huge paper

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang unThe Annals of Mathematics Second Series, Vol. 129, No. 1 (Jan., 1989), pp. 1-60

by Pierre Pansu. It was only luck that in 2000, at Lausanne, I met Sergey Vodop’yanov (from Sobolev Institute of Mathematics). He started to explain to me what Carnot groups are and I was thrilled to   learn that examples I needed previously are numerous in sub-riemannian geometry.

With the right frame of mind (at least I think so), that of intrinsic dilations, I  started then to study sub-riemannian geometry.

Planar rooted trees and Baker-Campbell-Hausdorff formula

Today on arXiv was posted the paper

with the abstract

We introduce the combinatorial notion of posetted trees and we use it in order to write an explicit expression of the Baker-Campbell-Hausdorff formula.

The paper may be relevant (check also the bibliography!) for the subject of writing “finitary“, “noncommutative” BCH formulae, from self-similarity arguments using dilations.

“Metric spaces with dilations”, the book draft updated

Here is the updated version.

Many things left to be done and to explain properly, like:

• the word tangent bundle and more flexible notions of smoothness, described a bit hermetically and by means of examples here, section 6,
• the relation between curvature and how to perform the Reidemeister 3 move, the story starts here in section 6 (coincidence)
• why the Baker-Campbell-Hausdorff formula can be deduced from self-similarity arguments (showing in particular that there is another interpretation of the Lie bracket than the usual one which says that the bracket is related to the commutator). This will be posted on arxiv soon. UPDATE:  see this post by Tao on the (commutative, say) BCH formula. It is commutative because of his “radial homogeneity” axiom in Theorem 1, which is equivalent with the “barycentric condition” (Af3) in Theorem 2.2 from “Infinitesimal affine geometry…” article.
• a gallery of emergent algebras, in particular you shall see what “spirals” are (a kind of generalization of rings, amazingly connecting by way of an example with another field of interests of mine, convex analysis).

Gleason metric and CC distance

In the series of posts on Hilbert’s fifth problem, Terence Tao defines a Gleason metric, definition 4 here, which is a very important ingredient of the proof of the solution to H5 problem.

Here is Remark 1. from the post:

The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

I want to explain why this is true. Look at the proof of theorem 7. The problem comes from the commutator estimate (1). I shall reproduce the relevant part of the proof because I don’t yet know how to write good-looking latex posts:

From the commutator estimate (1) and the triangle inequality we also obtain a conjugation estimate

$\displaystyle \| ghg^{-1} \| \sim \|h\|$

whenever ${\|g\|, \|h\| \leq \epsilon}$. Since left-invariance gives

$\displaystyle d(g,h) = \| g^{-1} h \|$

we then conclude an approximate right invariance

$\displaystyle d(gk,hk) \sim d(g,h)$

whenever ${\|g\|, \|h\|, \|k\| \leq \epsilon}$.

The conclusion is that the right translations in the group are Lipschitz (with respect to the Gleason metric). Because this distance (I use “distance” instead of “metric”) is also left invariant, it follows that left and right translations are Lipschitz.

Let now G be a connected Lie group with a left-invariant distribution, obtained by left translates of a vector space D included in the Lie algebra of G. The distribution is completely non-integrable if D generates the Lie algebra by using the + and Lie bracket operations. We put an euclidean norm on D and we get a CC distance on the group defined by: the CC distance between two elements of the group equals the infimum of lengths of horizontal (a.e. derivable, with the tangent in the distribution) curves joining the said points.

The remark 1 of Tao is a consequence of the following fact: if the CC distance is right invariant then D equals the Lie algebra of the group, therefore the distance is riemannian.

Here is why: in a sub-riemannian group (that is a group with a distribution and CC distance as explained previously) the left translations are Lipschitz (they are isometries) but not all right translations are Lipschitz, unless D equals the Lie algebra of G. Indeed, let us suppose that all right translations are Lipschitz. Then, by Margulis-Mostow version (see also this) of the Rademacher theorem , the right translation by an element “a” is Pansu derivable almost everywhere. It follows that the Pansu derivative of the right translation by “a” (in almost every point) preserves the distribution. A simple calculus based on invariance (truly, some explanations are needed here) shows that by consequence the adjoint action of “a” preserves D. Because “a” is arbitrary, this implies that D is an ideal of the Lie algebra. But D generates the Lie algebra, therefore D equals the Lie algebra of G.

If you know a shorter proof please let me know.

UPDATE: See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for details of the role of the Gleason metric  in the proof of the Hilbert 5th problem.

Hilbert fifth’s problem without one parameter subgroups

Further I reproduce, with small modifications, a comment   to the post

Locally compact groups with faithful finite-dimensional representations

by Terence Tao.

My motivation lies in the  project   described first time in public here.  In fact, one of the reasons to start this blog is to have a place where I can leisurely explain stuff.

Background:    The answer to the  Hilbert fifth’s problem  is: a connected locally compact group without small subgroups is a Lie group.

The key idea of the proof is to study the space of one parameter subgroups of the topological group. This space turns out to be a good model of the tangent space at the neutral element of the group (eventually) and the effort goes towards turning upside-down this fact, namely to prove that this space is a locally compact topological vector space and the “exponential map”  gives a chart  of  (a neighbourhood of the neutral element of ) the group into this space.

Because I am a fan of differential structures   (well, I think they are only the commutative, boring side of dilation structures  or here or emergent algebras)   I know a situation when one can prove the fact that a topological group is a Lie group without using the one parameter subgroups!

Here starts the original comment, slightly modified:

Contractive automorphisms may be as relevant as one-parameter subgroups for building a Lie group structure (or even more), as shown by the following result from E. Siebert, Contractive Automorphisms on Locally Compact Groups, Math. Z. 191, 73-90 (1986)

5.4. Proposition. For a locally compact group G the following assertions are equivalent:
(i) G admits a contractive automorphism group;
(ii) G is a simply connected Lie group whose Lie algebra g admits a positive graduation.

The corresponding result for local groups is proved in L. van den Dries, I. Goldbring, Locally Compact Contractive Local Groups, arXiv:0909.4565v2.

I used Siebert result for proving the Lie algebraic structure of the metric tangent space to a sub-riemannian manifold in M. Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136. arXiv:0804.0135v2

(added here: see  in Corollary 6.3 from “Infinitesimal affine …” paper, as well as Proposition 5.9 and Remark 5.10 from the paper  A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111 , arXiv:0810.5042v4 )

When saying that contractive automorphisms, or approximately contractive automorphisms [i.e. dilation structures], may be more relevant than one-parameter subgroups, I am thinking about sub-riemannian geometry again, where a one-parameter subgroup of a group, endowed with a left-invariant distribution and a corresponding Carnot-Caratheodory distance, is “smooth” (with respect to Pansu-type derivative) if and only if the generator is in the distribution. Metrically speaking, if the generator  is not in the distribution then any trajectory of the one-parameter group has Hausdorff dimension greater than one. That means lots of problems with the definition of the exponential and any reasoning based on differentiating flows.