A note on subgroups of automorphism groups of full shifts

TL;DR

This paper explores the structure of subgroups and submonoids within the automorphism and endomorphism groups of full shifts, revealing closure properties, embedding results, and undecidability aspects.

Contribution

It introduces new closure properties for subgroups and submonoids and demonstrates embeddings and undecidability results in the context of automorphism groups of full shifts.

Findings

Closure under direct products in endomorphism monoids

Closure under free products in automorphism groups

Embedding of automorphism groups into those of uncountable sofic shifts

Abstract

We discuss the set of subgroups of the automorphism group of a full shift, and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case, and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.