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

TL;DR

This paper explores the strong topological Rokhlin property (STRP) in countable groups, linking it to symbolic dynamics, shadowing, and group decompositions, and provides criteria and examples for when groups possess this property.

Contribution

It establishes that the STRP is a symbolic dynamical property, characterizes groups with STRP via density of certain subshifts, and connects STRP with shadowing, including applications to various classes of groups.

Findings

Groups decomposing as free products of finite or cyclic groups have the STRP.

Finitely generated nilpotent groups lack the STRP unless virtually cyclic.

Shadowing is generic for group actions if and only if the group has the STRP.

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 the STRP is a symbolic dynamical property. We prove that a countable group $G$ has the 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 the STRP. We show that finitely generated nilpotent groups do not have the 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 the STRP, this includes any group with infinitely generated center and the Hall universal locally finite group. We find a very strong connection between the STRP and shadowing, a.k.a. pseudo-orbit tracing property. We show that shadowing is generic for actions of a finitely generated group $G$ if and only if $G$ has the STRP.