# The Mapping Class Group of a Shift of Finite Type

## TL;DR

This paper investigates the structure and properties of the mapping class group associated with irreducible shifts of finite type, revealing its algebraic features and open questions in the area.

## Contribution

It characterizes the mapping class group of a shift of finite type, detailing its algebraic properties and establishing foundational results for flow equivalence.

## Key findings

- The group is countable and not residually finite.
- It acts n-transitively on circles in the mapping torus.
- The group has a solvable word problem and trivial center.

## Abstract

We study the mapping class group of a nontrivial irreducible shift of finite type: the group of flow equivalences of its mapping torus modulo isotopy. This group plays for flow equivalence the role that the automorphism group plays for conjugacy. It is countable; not residually finite; acts faithfully (and n-transitively, for all n) by permutations on the set of circles in the mapping torus; has solvable word problem and trivial center; etc. There are many open problems.