A characterization of high transitivity for groups acting on trees

A characterization of high transitivity for groups acting on trees, Discrete Analysis 2022:8, 63 pp.

Consider the group of all permutations of a countable set \(X\) that leave all but a finite number of points fixed. This is a countable group, and for each natural number \(n\) it also acts \(n\)-transitively on \(X\) in the following sense: given any two sequences \(x_1,\dots,x_n\) and \(y_1,\dots,y_n\), each consisting of \(n\) distinct elements of \(X\), there is an element of \(G\) that takes \(x_i\) to \(y_i\) for each \(i\). An action that is \(n\)-transitive for every \(n\) is called highly transitive. One then defines a group to be \(n\)-transitive if it admits an \(n\)-transitive faithful action, and highly transitive if it admits a highly transitive faithful action. (To see why the faithfulness condition is sensible, observe that if \(G\) admits a highly transitive action and \(H\) is any other group, then \(G\times H\) admits a highly transitive action, but one would like a definition that implies that the group in question has a very high degree of symmetry.)

An interesting open question, asked by Hull and Osin, is whether a group that is \(n\)-transitive for every \(n\) is automatically highly transitive. They asked the question in a paper that proved a central theorem in the area, which provided a large class of examples of highly transitive groups.

To understand their result, recall first the definition of a proper group action. The action of a group \(G\) on a metric space \(S\) is said to be proper if for any two compact sets \(K, K'\subset S\), the set of \(g\in G\) such that \(gK\cap K'\ne\emptyset\) is compact (and therefore finite if \(G\) is discrete).

Sometimes, one can get away with a variant of this property, which says that an action has a property that resembles properness except on a “thick diagonal” of \(S^2\) (that is, the set of pairs \((x,y)\in S^2\) such that \(x\) and \(y\) are “close”). An action is said to be acylindrical if for every \(t>0\) there exists \(R\) and \(N\) such that for every \(x,y\) with \(d(x,y)\geq R\), the number of \(g\in G\) such that \(d(x,gx)\leq t\) and \(d(y,gy)\leq t\) is at most \(N\). That is, there may be infinitely many \(g\) that map \(x\) to a point near \(x\), but if \(y\) is far from \(x\), then only finitely many of those \(g\) can send \(y\) to a point near \(y\).

A group is called acylindrically hyperbolic if it admits an acylindrical action on a hyperbolic metric space. This turns out to be a property with several equivalent definitions. It can also be shown that if \(G\) is acylindrically hyperbolic, then one can take the hyperbolic space on which \(G\) acts to be a Cayley graph of \(G\) with respect to a generating set, and that this space will be non-trivial in a certain sense. (That sense is that its “Gromov boundary” has size greater than 2, so the hyperbolic space doesn’t look like a copy of \(\mathbb R\).)

Hull and Osin proved that every acylindricaly hyperbolic group admits a highly transitive action with finite kernel. This hugely generalized many of the previous classes of examples that had been found and showed that high transitivity was a much wider phenomenon than had been previously thought. That said, it did not cover all known examples, and, as Hull and Osin pointed out, it also did not cover a class of groups that they believed to be highly transitive, namely (most of) the Baumslag-Solitar groups, which are groups with a presentation of the form \(\langle a,b:ba^mb^{-1}=a^n\rangle\). The Baumslag-Solitar groups were orginally defined as an example of non-Hopfian groups – that is, groups that are not isomorphic to any proper quotient of themselves – and have gone on to become an important source of examples for testing conjectures

This paper, as its title suggests, is concerned with examples of highly transitive groups that are not covered by previous results. The authors find a condition for certain groups acting on trees to be highly transitive. They demonstrate that their condition is sharp, in the sense that its hypotheses cannot be (sensibly) weakened. A sign that this result is a genuine advance is that it implies that the Baumslag-Solitar groups are highly transitive except when \(n=1, m=1\), or \(n=m\) (in which cases they can be shown not to be), thus answering the question of Hull and Osin.

The basic proof technique is interesting: it is to choose a suitable Polish space of group actions on a countable set \(X\), and to demonstrate that the set of faithful highly transitive actions is an intersection of dense open sets, and therefore is non-empty by the Baire category theorem. This technique was introduced by Dixon and used by three of the authors in an earlier paper, but here its application is more delicate and involved.