Vanishing of cohomology for groups acting on buildings

In his seminal paper from 1973, Garland introduced a machinery for proving vanishing of group cohomology for groups acting on Bruhat-Tits buildings. This machinery, known today as “Garland’s method”, had several applications as a tool for proving rigidity results (e.g., proving Kazhdan property (T) or, more recently, group stability results). In my talk, I will introduce a variation of Garland’s method, explain how it yields vanishing cohomology and how it can be generalized to vanishing of cohomology with Banach coefficients. Parts of this talk are based on joint works with Z. Grinbaum-Reizis and A. Lubotzky.