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

Abstract

We give a new proof, based on thermodynamic formalism, of a foundational result of Burger and Monod in bounded cohomology. Let $G$ be a noncompact connected semisimple real Lie group with finite center and no factors of real rank one, and let $\Gamma <G$ be a uniform lattice. We prove that, for every orthogonal representation $\pi :\Gamma \to O_N$, every $\pi$-quasimorphism $L:\Gamma \to {\mathbb{R}}^N$ is bounded.

Improved presentation and more generality in the main result. Some errors fixed