Presentations of Topological Full Groups by Generators and Relations

TL;DR

This paper characterizes the generators and relations for the commutator subgroup of topological full groups of minimal subshifts and links the solvability of the word problem to the recursiveness of the subshift's language.

Contribution

It provides a presentation of the commutator subgroup of topological full groups and establishes a criterion for the solvability of the word problem based on language recursiveness.

Findings

Generators and relations for the commutator subgroup are described.

Word problem solvability is equivalent to the language being recursive.

Provides a bridge between algebraic properties and symbolic dynamics.

Abstract

We describe generators and defining relations for the commutator subgroup of topological full groups of minimal subshifts. We show that the word problem in a topological full group is solvable if and only if the language of the underlying subshift is recursive.