Algebraic irrational stable commutator length in finitely presented groups

Abstract

We provide the first example of a finitely presented (in fact, type F∞) 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} Tτ: the circle analogue of the Cleary's golden ratio Thompson group Fτ which acts on the interval.