Unindexed subshifts of finite type and their connection to automorphisms of Thompson's groups

TL;DR

This paper introduces a new categorical framework generalizing subshifts of finite type, defining UDAF and weak UDAF equivalences, and connects these to automorphism groups of Higman, Thompson, and Brin groups.

Contribution

It proposes a novel category extending subshifts of finite type, introduces two new equivalence notions, and links these to automorphism groups of important algebraic structures.

Findings

UDAF equivalence does not imply shift equivalence.

UDAF equivalence equates the 2-leaf rose with the golden mean shift.

The category relates to automorphism groups of Higman, Thompson, and Brin groups.

Abstract

For a finite digraph $D$, we define the corresponding subshift of finite type $(X_D, \sigma_D)$ to be the dynamical system where $X_D$ is the set of all bi-infinite walks through $D$ and $\sigma_D$ is the shift operator. Two digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such that $(X_{D_1}, \sigma_{D_1}^n)$ and $(X_{D_2}, \sigma_{D_2}^n)$ are topologically conjugate for all $n\geq m$. They are called strong shift equivalent if this holds for $m=1$. In this paper we introduce a new category which generalises the category of subshifts of finite type and topological conjugacy. Our category gives two new notions of equivalence for digraphs which we call UDAF equivalence and weak UDAF equivalence. UDAF equivalence is a coarser analogue of strong shift equivalence and weak UDAF equivalence is a coarser analogue of shift equivalence. Both UDAF and weak UDAF equivalence still separate the n-leaf roses for all $n\geq 2$ (the 1-vertex n-edge digraphs). However, UDAF equivalence does not imply shift equivalence, in particular it equates the 2-leaf rose with the golden mean shift. We also show that our category relates to the automorphism groups of the "V-type" groups of Higman, Thompson and Brin. In particular for all $n\geq 2$ the groups $Out(G_{n, n-1})$ and $Out(nV)$ can be seen as automorphism groups of specific objects in our category. We explain a few equivalent ways of viewing UDAF equivalence, and give an example of how they can be used to show that Ashley's 8 vertex digraph is UDAF equivalent to the 2-leaf rose.