Groups acting on CAT(0) cube complexes with uniform exponential growth

Abstract

We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that non-virtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth.

Unbounded domains in hierarchically hyperbolic groups

“…For cubical groups, "many" means not virtually cyclic and not directly decomposable. Before this, the only results for cubical groups were those of Kar-Sageev in two dimensions [KS19], though these have since been extended to cube complexes with isolated flats by Gupta-Jankiewicz-Ng [GJN20]. (Again, it is interesting that in those settings the uniformity depends only on the dimension of the cube complex, and not on the choice of group.…”

Extensions of hyperbolic groups have locally uniform exponential growth

“…In this paper we study a variation of the Tits alternative, distinguishing between subgroups containing a uniformly short free basis, and those that satisfy a law. Recall that for a group G with finite generating set S and N ∈ N, a (free) subgroup H ≤ G is N -short (with respect to S) if there exists words of S-length at most N that generate H, and that G contains a uniformly N -short free subgroup if there exists an N -short free subgroup with respect to every finite generating set of G (see also [GJN20,Definition 1.1]). This definition motivates us to introduce the following property.…”

“…It is well-known that L Free ⊊ L UEG . Indeed, free Burnside groups of sufficiently large odd exponent [Osi07, Theorem 2.7] and solvable Baumslag-Solitar groups (see [GJN20,Lemma 6.3] and references therein) lie in L UEG but not L Free . Several of our statements apply to either of the above classes in which case we use the notation L * .…”