The BNSR-invariants of the Stein group 𝐹2,3

The Stein group F 2 , 3 F_{2,3} is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form 2 p ⁢ 3 q 2^{p}3^{q} ( p , q ∈ Z p,q\in\mathbb{Z} ) and breakpoints in Z ⁢ [ 1 6 ] \mathbb{Z}[\frac{1}{6}] . This is a natural relative of Thompson’s group 𝐹. In this paper, we compute the Bieri–Neumann–Strebel–Renz (BNSR) invariants Σ m ⁢ ( F 2 , 3 ) \Sigma^{m}(F_{2,3}) of the Stein group for all m ∈ N m\in\mathbb{N} . A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of F 2 , 3 F_{2,3} is of type F ∞ \operatorname{F}_{\infty} . Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map F 2 , 3 → Z F_{2,3}\to\mathbb{Z} is of type F ∞ \operatorname{F}_{\infty} , even though there exist maps F 2 , 3 → Z 2 F_{2,3}\to\mathbb{Z}^{2} whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ ∞ ⁢ ( F 2 , 3 ) \Sigma^{\infty}(F_{2,3}) , but there exist (non-discrete) characters that do not even lie in Σ 1 ⁢ ( F 2 , 3 ) \Sigma^{1}(F_{2,3}) . To the best of our knowledge, F 2 , 3 F_{2,3} is the first group whose BNSR-invariants are known exhibiting these properties.

On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

Non-triviality of the Poisson boundary of random walks on the group of Monod

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$ , we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.