Irrational Stable Commutator Length in Finitely Presented Groups

TL;DR

This paper demonstrates the existence of finitely presented groups with elements having irrational and transcendental stable commutator length, using examples from 1-dimensional dynamics related to generalized Thompson groups.

Contribution

It provides the first examples of such groups with irrational stable commutator length, answering a question posed by Gromov.

Findings

Existence of finitely presented groups with irrational stable commutator length

Examples derived from 1-dimensional dynamics and generalized Thompson groups

Answers Gromov's question negatively

Abstract

We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse and others.