Thermodynamic Formalism for Quasimorphisms: Lattices in Higher Rank Semisimple Lie Groups

Abstract

We give a proof, based on thermodynamic formalism, of a theorem in bounded cohomology extending a foundational result of Burger and Monod: if $\Gamma$ is an irreducible uniform lattice in a non-compact connected semisimple Lie group of real rank at least $2$, then for any finite-dimensional representation $\pi:\Gamma\to \operatorname{O}_N$, every $\pi$-quasimorphism (that is, a map with bounded defect with respect to $\pi$) is bounded.