Medvedev degrees of subshifts on groups

TL;DR

This paper explores the Medvedev degrees of subshifts on groups, analyzing how these degrees transfer across algebraic and geometric relations, and classifying their possible values on various groups.

Contribution

It develops a theory for transferring Medvedev degrees between groups and classifies degrees for subshifts of finite type on specific group classes.

Findings

Full classification of Medvedev degrees on virtually polycyclic groups

Classification of degrees for subshifts on branch groups with decidable word problem

Existence of SFTs with nonzero Medvedev degree on groups quasi-isometric to the hyperbolic plane

Abstract

The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be transferred from one group to another through algebraic and geometric relations, such as quotients, subgroups, translation-like actions and quasi-isometries. We use the aforementioned tools to study the possible values taken by this invariant on subshifts of finite type on some finitely generated groups. We obtain a full classification for some classes, such as virtually polycyclic groups and branch groups with decidable word problem. We also show that all groups which are quasi-isometric to the hyperbolic plane admit SFTs with nonzero Medvedev degree. Furthermore, we provide a classification of the degrees of sofic subshifts for several classes of groups.