Cogrowth for group actions with strongly contracting elements

Abstract

Let G be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let N be an infinite normal subgroup of G, and let δ N and δ G be the growth rates of N and G with respect to the pseudo-metric induced by the action. We prove that if G has purely exponential growth with respect to the pseudo-metric then δ N /δ G > 1/2. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk's original result on free groups with respect to a word metrics and a recent result of Jaerisch, Matsuzaki, and Yabuki on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

Patterson-Sullivan theory for groups with a strongly contracting element

“…Proposition 4.28 (Eymard [19,Exposé 1,§2]). Let H be a subgroup of G such that H is co-amenable in G. Let V be a locally convex, topological vector space, endowed with a continuous affine action of G. Let K be a G-invariant, compact, convex subset of V .…”

“…Some of our results refine existing statements in the literature. In particular, we answer most of the questions raised by Arzhantseva and Cashen in [2]. Our main contribution though is the method that we use: we extend to this context the construction of Patterson-Sullivan measures (see below).…”

“…is bounded from above and away from zero [2]. Note that even if X is Gromov hyperbolic, there are groups G acting on X, which are divergent but do not have pure exponential growth.…”

“…In the past years, growth problems in groups with a contracting element have been investigated by various people, see for instance [46,1,17,47,48,2,29].…”

Critical exponents of invariant random subgroups in negative curvature

“…This was later generalized by Matsuzaki-Yabuki-Jaerisch [MYJ15] to deal with normal subgroups of discrete isometry groups of proper Gromov hyperbolic spaces. Arzhantseva and Cashen [AC18] have recently generalized [MYJ15] to normal subgroups of finitely generated groups acting on proper geodesic spaces with a strongly contracting element. This class includes rank-one actions on CAT(0) spaces as well as the mapping class group action on the Teichmüller space equipped with the Teichmüller metric.…”