An Irrational-slope Thompson's Group
Jos\'e Burillo, Brita Nucinkis, Lawrence Reeves

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 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 in are undistorted.
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Taxonomy
Topicssemigroups and automata theory · Geometric and Algebraic Topology · Advanced Combinatorial Mathematics
