Embeddings into Thompson's groups from quasi-median geometry

TL;DR

This paper demonstrates that various braided diagram groups can be embedded into Thompson's groups by analyzing their quasi-median geometry, revealing new structural relationships and embedding results.

Contribution

It establishes a splitting theorem for braided diagram groups into subgroups of right-angled Artin groups and Thompson's groups, and shows embeddings of several important groups into Thompson's group V.

Findings

Braided diagram groups split into subgroups of right-angled Artin groups and Thompson's groups.

Several groups, including Higman's groups and Houghton's groups, embed into Thompson's group V.

The approach uses quasi-median geometry to analyze group embeddings.

Abstract

The main result of this article is that any braided (resp. annular, planar) diagram group $D$ splits as a short exact sequence $1 \to R \to D \to S \to 1$ where $R$ is a subgroup of some right-angled Artin group and $S$ a subgroup of Thompson's group $V$ (resp. $T$, $F$). As an application, we show that several braided diagram groups embeds into Thompson's group $V$, including Higman's groups $V_{n,r}$, groups of quasi-automorphisms $QV_{n,r,p}$, and generalised Houghton's groups $H_{n,p}$.