Tag Archives: topological groups

June 15, 2013 · 9:31 pm

This paper relates two important invariants associated to orbit equivalence relations. So let’s define what types of equivalence relations we’re talking about (and a related group), what the two invariants are, and then we’ll say what the main result of the paper actually is.

Suppose we have a countable group $\Gamma$ acting on a probability space $(X,\mu)$ . Then the action is measure-preserving, well, if it preserves the measure. So $\mu(g\cdot B)=\mu(B)$ for any $g\in\Gamma$ and measurable $B\subseteq X$ . The orbit equivalence relation $E_\Gamma$ on $X$ is defined by

$x E_\Gamma y \Leftrightarrow \exists g\in\Gamma (g\cdot x = y)$

We call the equivalence relation probability measure-preserving if it arises from a measure-preserving action. We call it ergodic if every measurable union of $E_\Gamma$ -classes is either measure 0 or measure 1. This means, for instance, that if $C\subseteq X$ has positive measure and we look at $O=\cup_{g\in\Gamma} g\cdot C$ (i.e. the union of the $E_\Gamma$ -classes which meet $C$ ), then $\mu(O)=1$ . So in some sense, group actions which give rise to ergodic equivalence relations (unsurprisingly called ergodic group actions) must move things around a lot, since sets of positive measure will cover the whole space (except for a null set) when you act on them by the group. Of course, there are stronger mixing properties out there, but that doesn’t concern us presently.

Given a probability measure preserving equivalence relation $E$ , one can talk about all measure-preserving automorphisms $T$ of $(X,\mu)$ for which $T(x) E x$ for $\mu-a.e.$ $x\in X$ . This forms a group $[E]$ , called the full group of $E$ . Note that if $E=E_\Gamma$ for $\Gamma$ acting on $X$ in a measure-preserving way, then $\Gamma\leq [E]$ .

The full group of $E$ can be turned into a topological group without too much trouble (the metric is the obvious one, where the distance between two functions is the measure of the elements on which they differ), and then we can bring topological group theory to bear when we talk about it. The structure of $[E]$ is related to properties of $E$ in ways that we’re still exploring. One fact is that $[E]$ is topologically simple (i.e. it has no nontrivial closed normal subgroups) iff $E$ is ergodic.

The invariants in question

Suppose we have a measure-preserving equivalence relation $E$ . We extend our viewpoint beyond even $[E]$ to look at partial Borel automorphisms of $(X,\mu, E)$ . By this I mean Borel bijections of the form $f\colon A\rightarrow B$ , where $A,B\subseteq X$ are Borel, $f(x) E x$ for $\mu-a.e.$ $x\in X$ , and $f$ preserves the measures induced on $A$ and $B$ . We write this set as $[[E]]$ .

So suppose we have a collection $F=\{f_1,f_2,\ldots\}\subseteq [[E]]$ . We can define a graph $G_F$ by $x G_F y \Leftrightarrow \exists f_i\in F (f^{\pm 1}(x)=y)$ . If the connected components of $G_F$ are the equivalence classes of $E$ we call $F$ a graphing of $E$ . Then the cost of $F$ is

$\sum_i \mu(\text{dom}(f_i))$ ,

which one can check is the same as the average degree of a vertex in $G_F$ . The cost of $E$ is the infimum of the costs of its graphings. We’ll write it $C_\mu(E)$ . (The standard reference for this stuff, by the way, is Topics in Orbit Equivalence, by Kechris and Miller.)

The other invariant we’ll look at is the number of topological generators of $[E]$ . We say $\{g_1, g_2,\ldots\}\subseteq [E]$ is a set of topological generators of $[E]$ if the group generated by $\{g_1,g_2,\ldots\}$ is dense in $[E]$ . We write $t([E])$ for the least number of elements in a set of topological generators for $[E]$ .

The main theorem

If $E$ is a probability measure-preserving ergodic equivalence relation, then $t([E])=\lfloor C_\mu(E) \rfloor + 1$ .

The proof itself is fairly short, although it does rely on some earlier results on cost and full groups. Really, this is as nice a relationship as one could hope for. The author mentions that they are currently working on a paper in the case of non-ergodic equivalence relations, and have some similar results there.

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April 22, 2013 · 6:04 pm

(Non)-Topologizable groups

Last post was somewhat lengthy, so I’ve decided to just cover one paper from the ArXiv this time. Also, I’ve decided to change up my post-naming convention, so that it will hopefully be a bit more useful.

On topologizable and non-topologizable groups

A.A. Klyachko (Moscow State University), A. Yu. Olshanskii (Moscow State University and Vanderbilt University), Denis Osin (Vanderbilt University)

A group is called topologizable if it admits a non-discrete Hausdorff group topology. This paper is interested in several questions around this notion; the main focus of the paper seems to be the notion of c-compactness, an analog for topological groups of compactness of topological spaces. The main result of the paper is that these two notions do not coincide.

The proof uses non-topologizable groups (in fact, it uses a stronger property). What’s more interesting to me is the material in the second half of the paper regarding topologizable groups. Some background is needed…

Back in 1985, Grigorchuk introduced a topology on the space of finitely generated groups. It’s easier to describe the topology on the space $\mathcal G_k$ of $k$ -generated groups for some $k\in\mathbb{N}$ , and in practice this is often what one works with. (For the full space of finitely generated groups, take the countable union of the $\mathcal G_k$ . Yes, there is a sensible way in which $\mathcal G_k\subseteq \mathcal G_{k+1}$ .) There are two equivalent ways to define it:

Let $\mathbb{F}_k$ denote the free group on $k$ generators. Then every $k$ -generated group is a homomorphic image of $\mathbb{F}_k$ , and so can be identified with a normal subgroup of $\mathbb{F}_k$ . The collection of subsets of $\mathbb{F}_k$ has a natural topology; identifying a subset of $\mathbb{F}_k$ with its characteristic function gives an element of $2^{\mathbb{F}_k}$ , and this can be given the product topology. The space of normal subgroups of $\mathbb{F}_k$ is simply given the subspace topology.
Alternately, let $g_1,\ldots, g_k$ be a set of generators for a $k$ -generator group $G$ . We call the ordered pair $(G, (g_1,\ldots, g_k))$ a ( $k$ -)marked group. Then one may define an associated Cayley graph in the usual way; the vertices are the elements of $G$ , and $g$ and $h$ are connected if $g=hg_i^{\pm 1}$ for some $1\leq i\leq k$ . Then two marked groups are close together if their Cayley graphs agree for a large ball around the identity (there are many equivalent metrics which capture this idea, just pick one that suits you).

To see that these are really the same thing, identify a marked group $(G, (g_1,\ldots, g_k))$ with the kernel of the map from $\mathbb{F}_k$ to $G$ sending $x_i$ to $g_i$ for $1\leq i\leq n$ . I like the second picture more, although I’ll admit that it is often easier to verify the topological properties of sets in this space using the first. It’s not too hard to see that this space is compact and completely metrizable, meaning it’s quite pleasant to work with in some sense. Of course, it’s totally disconnected, but as a descriptive set theorist this likely bothers me less than it should.

The authors show that the set of topologizable groups is $G_\delta$ and so is the set of Tarski monsters. (Recall that a group is called a Tarski monster if it is an infinite simple group with every subgroup finite cyclic. Truly bizarre creatures.) They then show that there is a subset $\mathcal T\subseteq \mathcal G_2$ which is completely metrizable and in which both of these two sets are dense. Thus by the Baire Category Theorem there exists a dense (in $\mathcal T$ ) $G_\delta$ set of topologizable Tarski monsters. And yet they never produce a single one! You can find this admirable or aggravating, but I tend to lean toward the former.

The construction of these dense subsets relies heavily on the results of Olshanskii’s book Geometry of Defining Relations in Groups, which shouldn’t be too surprising as this is where the first Tarski monsters were constructed. I’m not too familiar with the techniques (beyond the fact that they involve generalizing small cancellation theory in some sense), but my understanding is that the book is quite readable. It’s on my “to-read” list, but unfortunately not anytime soon.

Some ArXiv findings, Mar. 29

Let’s take another look at the ArXiv, shall we? (Originally the plan was for this to be weekly, oops.)

On embeddings into compactly generated groups

Pierre-Emmanuel Caprace (Université catholique de Louvain) and Yves Cornulier (University of Paris-South (Orsay))

One of the great triumphs of mid-20th-century combinatorial group theory was the result (due to Higman, Neumann, and Neumann) that any countable group can embed into a 2-generator group. This paper starts by mentioning this result and several other embedding results in the same vein.

[As an aside, in the introduction the authors state of the above theorem, “This was a major breakthrough, providing some of the first evidence that finitely generated groups are not structurally simpler than countable groups and thus are far from tame or classifiable.” Descriptive set theory actually gives us a framework to discuss exactly these issues, and in this framework the above sentence is half-right. One speaks of the complexity of the equivalence relation associated to a classification problem, and two different equivalence relations are compared using Borel reductions. These slides (pdf) give an overview.

Presumably here one would like to talk about classifying groups up to isomorphism. It’s true that classifying finitely generated groups up to isomorphism is far from tame; indeed isomorphism of finitely generated groups is what we call a universal countable Borel equivalence relation, meaning it’s as complicated as it could possibly be. However, isomorphism of countable groups is a universal among all equivalence relations arising from a (Polish, i.e. continuous) $S_\infty$ -action. This is far more complicated than universal countable. So it would seem safe to say that finitely generated groups are in fact structurally simpler than countable groups, although they are still not structurally simple in absolute terms.

If instead we were to look at the biembeddability relation (the equivalence relation arising naturally from the embeddability structure), then the gap is even larger. Biembeddability of finitely generated groups is a universal countable Borel equivalence relation, but biembeddability of countable groups is a universal analytic equivalence relation. This (largely expository) article from the Notices has a picture of how these are related.]

All of these results are in the context of countable discrete groups. As of late, there has been a renewed interest in topological groups (previously, much of the focus here was on Lie groups), and so it makes sense to ask if there are results of a similar flavor to be found here. Natural analogs of “countable” and “finitely generated” are “ $\sigma$ -compact” and “compactly generated”. One might hope for a theorem like that of Higman, Neumann, and Neumann, with these terms in place of the originals. An old result of Pestov shows that this holds for topological Hausdorff groups. The authors show that this is not possible when looking at locally compact groups:

Theorem [Caprace, Cornulier]: There exists a second countable (hence $\sigma$ -compact), topologically simple totally disconnected locally compact group S, such that every continuous (or even abstract) homomorphism of S to any compactly generated locally compact group is trivial.

There are some sharper results in the paper, and a positive result regarding embedding abelian groups. I’m particularly interested in ruling out some group embedding into entire classes of other groups, as I’m currently interested in the question of whether every group embeds into a co-Hopfian group. (See here.) In this paper they manage to relate the fact that a group H has no non-trivial continuous actions on a graph of bounded degree to the fact that all of its continuous homomorphisms to compactly generated groups are trivial. (This is not an equivalence, by the way.) Could there be something similar when talking about embeddings of countable groups into co-Hopfian groups?

An uncountable family of 3-generated groups with isomorphic profinite completions

Volodymyr Nekrashevych (Texas A&M)

The title of this paper brings to mind a previous paper of Nekrashevych, “A minimal Cantor set in the space of 3-generated groups”. In that paper, he defines a family of branch, just-infinite finitely generated groups for which isomorphism is not smooth (in the sense of Borel equivalence relations). This is notable, because there are very few constructions of this sort. Generally speaking, when group theorists wish to show a given class of groups is large, they perform a smooth construction of continuum-many distinct-up-to-isomorphism groups. But as mentioned earlier, isomorphism of finitely generated groups is rather complicated, and so such constructions could in theory be much more complex. Constructions using free products with amalgamation are currently the only ones we know of which achieve the full complexity. Free products with amalgamation let you do all sorts of things, so the question is if there are any other natural classes of groups for which isomorphism is this complex, for example amenable groups. The difficulty is that one is more restricted in the sorts of constructions one can do while staying in such classes. A first step is to show that isomorphism for these classes is not smooth, as Nekrashevych did for just-infinite branch groups.

This paper uses the same family of groups. The main result is that the members of this family are residually finite and have the same profinite completion. A few other properties of the family are also mentioned. This paper stands alone, and so there is significant overlap with the one I was discussing last paragraph. I imagine that the exposition of the construction might be more polished, simply due to the additional time Nekrashevych has spent thinking about it, but I’ll admit to not having read either paper in depth at this point.

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Some ArXiv findings, Mar. 29

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