(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.


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