# Topology of the double of the double of the double …

In Sub-riemannian geometry and Lie groups , Part I, section 4.2, I introduced the strange notion of “uniform group”. Instead of saying that an uniform group is just a topological group (which has an unique uniformity associated to it), I proposed the following construction.

1. Double of a group.  To any group $G$ we associate its double group $G^{(2)} = G \times G$ with the group operation

$(x,u) \, (y,v) = (xy, y^{-1}u y v)$

I introduce also the following three functions:

$op: G^{(2)} \rightarrow G$ , $op(x,u) = xu$,

$i': G \rightarrow G^{(2)}$ , $i'(x) = (x,e)$   (where $e$ is the neutral element of the group $G$),

$i": G \rightarrow G^{(2)}$ , $i"(x) = (x, x^{-1})$.

Remark that all these functions are group morphisms. Notice especially the morphism $op$, which is nothing but the group operation of $G$, seen as a group morphism from $G^{(2)}$ to $G$.

2. Uniform group. For my purposes I introduced the notion of an uniform group: an uniform group is a group $G$, together with two uniformities, one on $G$, the other on $G^{(2)}$, such that the three morphisms from the point 1. are uniformly continuous.

So, instead of one uniformity, now I use two. Why? Well, we may eliminate one of the uniformities, the one on $G$. Indeed, suppose that  $G^{(2)}$  is an uniform group. Then take on $G$ the smallest uniformity which makes the three morphisms uniformly continuous. Now we have two uniformities, one on the double of $G$, the other on the double of the double of $G$. We may repeat the procedure indefinitely, pushing to infinity the pair of uniformities.

Wait, what?

(TO BE CONTINUED)

# Uniform spaces, coarse spaces, dilation spaces (II)

Background:
(1) Uniform spaces, coarse spaces, dilation spaces

I shall use the idea explained in (1), in the groupoid frame of (2), classical situation of the trivial groupoid over a space $X$. In this case the uniform and coarse structures are just the classical notions.

This idea says that all we need to have is a field of dilations. With such an object we may construct an uniformity, or a coarse structure, then ask that the field of dilations has some properties with respect to the said uniformity (or coarse structure). If the field of dilations has those properties then it transforms into an emergent algebra  (in the case $0$ below; in the other case  there is a new type of emergent algebra which appears in relation to coarse structures).

Remark. Measure can be constructed from a field of dilations, that’s for later.

Fields of dilations. We have a set $X$, call it “space”. We have the commutative group $(0,\infty)$ with multiplication of reals operation, (other groups work well, let’s concentrate on this one).

A field of dilations is a function which associates to any $x \in X$ and any $\varepsilon \in (0,\infty)$
an invertible transformation  $\delta^{x}_{\varepsilon} : U(x) \subset X \rightarrow U(x,\varepsilon) \subset X$ which we call “the dilation based at $x$, of coefficient $\varepsilon$“.

1. $x \in U(x)$ for any point $x \in X$

2. for any fixed $x \in X$ the function $\varepsilon \in (0,\infty) \rightarrow \delta^{x}_{\varepsilon}$ is a representation of the commutative group $(0,\infty)$.

3. fields of dilations come into 2 flavors (are there more?), depending on the choice between $(0,1]$ and $[1,\infty)$, two important sub-semigroups of $(0,\infty)$.

(case $0$) – If you choose $(0,1]$ then we ask that for any $\varepsilon \in (0,1]$ we have $U(x,\varepsilon) \subset U(x)$, for any $x$,

This case is good for generating uniformities and  for the infinitesimal point of view.

(case $\infty$) – If your choice is $[1,\infty)$ then we ask that for any $\varepsilon \in [1,\infty)$ we have $U(x) \subset U(x,\varepsilon)$, for any $x$,

This case is good for generating coarse structures and for  the asymptotic point of view.

Starting from here, I’m afraid that my latex capabilities on wordpress are below what I need to continue.

PS. At some point, at least for the case of uniformities, I shall use “uniform refinements” and what I call “topological derivative” from arXiv:0911.4619, which can be applied for giving alternate proofs for rigidity results, without using Pansu’s Rademacher theorem in Carnot groups.

# Uniform spaces, coarse spaces, dilation spaces

This post is related to the previous What’s the difference between uniform and coarse structures? (I).  However, it’s not part (II).

Psychological motivation.  Helge Gloecker made a clear mathscinet review of my paper Braided spaces with dilations and sub-riemannian symmetric spaces.  This part of the review motivated me to start writing some more about dilatation structures, some of it explained in the next section of this post. Here is the motivating part of the review:

“Aiming to put sub-Riemannian geometry on a purely metric footing (without recourse to differentiable structures), the author developed the concept of a dilatation structure and related notions in earlier works [see J. Gen. Lie Theory Appl. 1 (2007), no. 2, 65–95; MR2300069 (2008d:20074); Houston J. Math. 36 (2010), no. 1, 91–136; MR2610783 (2011h:53032); Commun. Math. Anal. 11 (2011), no. 2, 70–111; MR2780883 (2012e:53051)].

[…]

In a recent preprint [see “Emergent algebras”, arXiv:0907.1520], the author showed that idempotent right quasigroups are a useful auxiliary notion for the introduction (and applications) of dilatation structures.

[…]

The author sketches how dilatation structures can be defined with the help of uniform irqs, suppressing the technical and delicate axioms concerning the domains of the locally defined maps. More precisely, because the maps are only locally defined, the structures which occur are not uniform irqs at face value, but only suitable analogues using locally defined maps (a technical complication which is not mentioned in the article).”

This was done before, several times, for dilatation structures, but Helge is right that it has not been done for emergent algebras.

Which brings me to the main point: I started a paper with the same (but temporary) name as this post.  The paper is based on the following ideas.

Fields of dilations which generate their own topology, uniformity, coarse structure, whatever.    In a normed real vector space $(V, \| \cdot \|)$ we may use dilations as replacements of metric balls. Here is the simple idea.

Let us define, for any $x \in V$, the domain of $x$ as $U(x)$,  the ball with center $x$, of radius one, with respect to the distance $d(x,y) = \|x - y \|$. In general, let $B(x, r)$  be the  metric ball with respect to the distance induced by the norm.

Also, for any $x \in V$ and $\varepsilon \in (0,+\infty)$, the dilation based at $x$, with coefficient $\varepsilon$ is the function $\delta^{x}_{\varepsilon} y = x + \varepsilon ( -x + y)$ .

I use these notations to write the  ball of radius $\varepsilon \in (0,1]$ as $B(x,\varepsilon) = \delta^{x}_{\varepsilon} U(x)$ and give it the fancy name dilation ball of coefficient $\varepsilon$.

In fact, spaces with dilations, or dilatation structures, or dilation structures, are various names for spaces endowed with fields of dilations which satisfy certain axioms. Real normed vector spaces are examples of spaces with dilations, as a subclass of the larger class of conical groups with dilations, itself just a subclass of groups with dilations. Regular sub-riemannian manifolds are examples of spaces with dilations without a predefined group structure.

For any space with dilations, then, we may ask what can we get by forgetting the distance function, but keeping the dilation balls.  With the help of those we may define the uniformity associated with a field of dilations and then ask that the field of dilations is behaves uniformly, and so on.

Another structure, as interesting as the uniformity structure, is the (bounded metric) coarse structure of the space, which  again could be expressed in terms of fields of dilations. As coarse structures and uniform structures are very much alike (only that one is interesting for the small scale, other for the large scale), is there a notion of dilation structure which is appropriate for coarse structures?

I shall be back on this, with details, but the overall idea is that a field of dilations (a notion even weaker than an emergent algebra) is somehow midway between a uniformity structure and a coarse structure.

UPDATE (28.10.2012): I just found bits of the work of Protasov, who introduced the notion of “ball structure” and “ballean”, which is clearly related with the idea of keeping the balls, not the distance. Unfortunately, for the moment I am unable to find a way to read his two monographs on the subject.

# 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.