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

We give a proof, based on thermodynamic formalism, of a theorem in bounded cohomology extending a foundational result of Burger and Monod: if $\mathrm{\Gamma }$ 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 $\pi :\mathrm{\Gamma }\to {\mathrm{O}}_N$, every $\pi$-quasimorphism (that is, a map with bounded defect with respect to $\pi$) is bounded.