Algebraic irrational stable commutator length in finitely presented groups

TL;DR

This paper constructs the first example of a finitely presented group with elements having algebraic irrational stable commutator length, addressing a question in geometric group theory.

Contribution

It introduces the golden ratio Thompson group $T_ au$ and demonstrates its elements have algebraic irrational stable commutator length, a novel example in the field.

Findings

First example of finitely presented group with algebraic irrational stable commutator length

The group $T_ au$ acts on the circle and lifts to the real line

Answers a longstanding question by Calegari

Abstract

We provide the first example of a finitely presented (in fact, type $F_\infty$) group with elements whose stable commutator length is algebraic and irrational, answering a question of Calegari. Our example is the lift to the real line of the \emph{golden ratio Thompson group} $T_\tau$: the circle analogue of the Cleary's golden ratio Thompson group $F_\tau$ which acts on the interval.