Thompson's $V$ in MCG of mixing SFT by PW-linear homeos

TL;DR

This paper demonstrates how Thompson's group V can be embedded into the mapping class group of certain subshifts using piecewise linear local rules, revealing a new connection between these groups.

Contribution

It provides an explicit construction of the embedding of Thompson's V into the mapping class group via piecewise linear local rules, extending previous theoretical results.

Findings

Thompson's V embeds in the mapping class group of certain subshifts.

The embedding is achieved through piecewise linear local rules.

The embedding splits into the mapping class group via homeomorphisms.

Abstract

In a recent paper, we showed that groups admitting "veelike actions" on a finite language embed in mapping class groups of certain two-sided subshifts. In this note, we illustrate this theorem for the embedding of Thompson's $V$ by exhibiting the piecewise linear local rules for the embedding. These turn out to split the embedding into the mapping class group, showing that $V$ even embeds in the mapping class group by homeomorphisms.