On Volume Rigidity of Lattices

Our goal in this Master’s thesis is to give a detailed proof of the volume rigidity theorem due to Bucher, Burger, and Iozzi, following the lines of the article.

Along the way, background information on hyperbolic geometry and in particular on continuous (bounded) cohomology is provided, introducing the reader to the subject. We also prove a version of de Rham’s theorem for relative de Rham cohomology in the appendix. Further a detailed discussion of Douady-Earle’s barycenter construction for probability measures on \(\partial \mathbb{H}^n\) with no atoms of mass ≥ 1/2 is included.