The strong topological Rokhlin property and Medvedev degrees of SFTs

TL;DR

This work identifies a practical sufficient condition for the failure of the strong topological Rokhlin property: a recursively presented group that admits a nonempty subshift of finite type with nonzero Medvedev degree cannot have STRP. The authors leverage Doucha’s criterion, projectively isolated subshifts, and computability properties of subshifts to show that such an SFT forces neighborhoods without projectively isolated subshifts, thereby obstructing STRP. The result yields new examples of recursively presented groups without STRP and clarifies the computability-dynamical relationship central to the STRP problem. The approach provides a unifying, simpler criterion that subsumes previous conditions and opens questions about relativized Medvedev degrees and broader group classes.

Abstract

We prove that if a recursively presented group admits a (nonempty) subshift of finite type with nonzero Medvedev degree then it fails to have the strong topological Rokhlin property. This result simplifies a known criterion and provides new examples of recursively presented groups without this property.

Theorems & Definitions (15)

• Theorem 1.1
• Remark 2.1
• Definition 2.2: doucha_strong_2024
• Remark 2.3
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof