Automorphism groups of random substitution subshifts

TL;DR

This paper investigates the automorphism groups of random substitution subshifts, revealing they contain complex structures like infinite simple groups and automorphisms of full shifts, and introduces new concepts like shuffles and recognisability.

Contribution

It introduces the concepts of shuffles, generalized shuffles, and a local recognisability condition for random substitutions, advancing understanding of their automorphism groups.

Findings

Automorphism groups contain an infinite simple subgroup.

Automorphism groups include a copy of the automorphism group of a full shift.

Subshifts are countable, non-amenable, and non-residually finite.

Abstract

We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.