Undecidable word problem in subshift automorphism groups

TL;DR

This paper explores the complexity of the word problem in automorphism groups of subshifts, demonstrating that for any Turing degree, there exists a subshift with an automorphism subgroup matching that degree's complexity.

Contribution

It establishes the existence of subshifts with automorphism groups whose word problems can have any prescribed Turing degree, highlighting the rich complexity spectrum.

Findings

Automorphism groups can have word problems of arbitrary Turing degrees.

Existence of subshifts with automorphism groups matching any Turing degree.

Complexity of the word problem varies widely across subshift automorphism groups.

Abstract

This article studies the complexity of the word problem in groups of automorphisms of subshifts. We show in particular that for any Turing degree, there exists a subshift whose automorphism group contains a subgroup whose word problem has exactly this degree.