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 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 ) 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 (and use the relations between generators), then we use the inverse dilation (which is a dilation of coefficient ) to obtain the “approximate sum” of the two elements!
In practice, we probably have a dilation of coefficient which could simplify the computation of products of group elements of length at most, for example.
But it looks like a solution to the problem of not getting bored, at least to me.