Automata groups generated by Cayley machines of groups of nilpotency class two

We build presentations for automata groups generated by Cayley machines of finite groups of nilpotency class two and prove that these automata groups are all cross-wired lamplighter groups.

Hyperbolic structures on wreath products

The study of the poset of hyperbolic structures on a group G was initiated by C. Abbott, S. Balasubramanya and D. Osin, [Hyperbolic structures on groups, Algebr. Geom. Topol. 19 2019, 4, 1747–1835, 10.2140/agt.2019.19.1747]. However, this poset is still very far from being understood, and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the lamplighter groups {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} all have finitely many quasi-parabolic structures.

The degree of commutativity and lamplighter groups

The degree of commutativity of a group [Formula: see text] measures the probability of choosing two elements in [Formula: see text] which commute. There are many results studying this for finite groups. In [Y. Antolín, A. Martino and E. Ventura, Degree of commutativity of infinite groups, Proc. Amer. Math. Soc. 145 (2017) 479–485, MR 3577854], this was generalized to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form [Formula: see text] and [Formula: see text], where [Formula: see text] is any finite group.

Furstenberg entropy of intersectional invariant random subgroups

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.