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

Michal Doucha

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.

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

Abstract

A countable group 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 has the STRP if and only if certain sofic subshifts over are dense in the space of subshifts. A sufficient condition is that isolated shifts over 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 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 if and only if has the STRP.
Paper Structure (15 sections, 3 theorems, 101 equations)

This paper contains 15 sections, 3 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Cantor and symbolic dynamics
  3. Spaces of actions and shifts
  4. Spaces of actions
  5. Subshits and their spaces
  6. Isolated subshifts, subshifts of finite type, and sofic subshifts
  7. Strong topological Rokhlin property
  8. Failure of the strong topological Rokhlin property
  9. Establishing the strong topological Rokhlin property
  10. Free products with the strong topological Rokhlin property
  11. Groups without the strong topological Rokhlin property
  12. Finitely generated groups without the STRP
  13. STRP for groups that are not finitely generated
  14. Shadowing
  15. Remarks, problems, and questions

Key Result

Theorem A

Let $G$ be a countable group. If $G$ is finitely generated, then the following are equivalent. For a general countable $G$, we have the equivalence between it:intro1-1 and it:intro1-2, and moreover it:intro1-3 implies it:intro1-1 and it:intro1-2.

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Theorem C
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 24 more