An Irrational-slope Thompson's Group

TL;DR

This paper explores the properties of the irrational-slope Thompson's group $F_\tau$, including its presentations, combinatorial structure, simplicity of its commutator subgroup, and a normal form for its elements, revealing geometric embedding properties.

Contribution

It introduces a detailed study of $F_\tau$, including new presentations, a normal form, and analysis of its geometric and algebraic properties, extending understanding of Thompson's groups.

Findings

The commutator subgroup of $F_\tau$ is simple.

Several embeddings of $F$ into $F_\tau$ are undistorted.

A unique normal form for elements of $F_\tau$ is established.

Abstract

The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_\tau$ introduced by Cleary in 1995. We construct presentations, both finite and infinite and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of $F$ in $F_\tau$ are undistorted.