Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups

Abstract

A countable group $G$ has the strong topological Rokhlin property (STRP) if it admits a continuous action on the Cantor space with a comeager conjugacy class. We show that having STRP is a symbolic dynamical property. We prove that a countable group $G$ has STRP if and only if certain sofic subshifts over $G$ are dense in the space of subshifts. A sufficient condition is that isolated shifts over $G$ are dense in the space of all subshifts.
We provide numerous applications including the proof that a group that decomposes as a free product of finite or cyclic groups has STRP. We show that finitely generated nilpotent groups do not have STRP unless they are virtually cyclic; the same is true for many groups of the form $G_1\times G_2\times G_3$ where each factor is recursively presented. We show that a large class of non-finitely-generated groups do not have STRP; this includes any group with infinitely generated center and the Hall universal locally finite group.
We find a very strong connection between STRP and shadowing, also called the pseudo-orbit tracing property. We show that shadowing is generic for actions of a finitely generated group $G$ if and only if $G$ has STRP.