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

TL;DR

This paper applies thermodynamic formalism to bounded cohomology, proving that certain quasimorphisms on lattices in higher rank semisimple Lie groups are bounded, extending foundational results.

Contribution

It introduces a thermodynamic formalism approach to bounded cohomology, providing a new proof of boundedness of quasimorphisms in higher rank lattices.

Findings

Every π-quasimorphism on such lattices is bounded.

The proof extends a result of Burger and Monod.

Thermodynamic formalism is effective in bounded cohomology.

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.