Thompson's Group F

TL;DR

This paper introduces forest and strand diagrams for Thompson's group F, providing new tools for understanding its structure, length formula, and applications to its geometric and algebraic properties, including presentations and classifying spaces.

Contribution

The paper develops forest and strand diagrams for F, derives a length formula, and constructs a classifying space, offering new insights and simplified proofs for existing properties.

Findings

Derived a simple length formula for F

Constructed a classifying space for F as a configuration space

Presented new bounds on the isoperimetric constant of F

Abstract

We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for braid groups, we present a classifying space for F that is the ``configuration space'' of finitely many points on a line, with the points allowed to split and merge in pairs. Strand diagrams are related to a description of F as a groupoid, which we use to derive presentations for F, T, V, and the braided Thompson group BV. In addition to the new results, we include a thorough exposition of the basic theory of the group F. Highlights include a simplified proof that the commutator subgroup of F is simple, a discussion of open problems (with a focus on amenability), and a simplified derivation of the standard presentation and normal forms for F using forest diagrams.