Date: Wednesday, 14 September, 2022.
Time: 3:30 PM.
Venue: Seminar Hall
Cohomology of groups and spaces: locally symmetric spaces and higher expanders
Chennai Mathematical Institute.
We will see two analogous situations in which cohomology of spaces can be expressed as that of groups (or related objects), leading to a group theoretic approach to their study. The first is de Rham cohomology of (compact) locally symmetric spaces and the second is simplicial cohomology of a finite quotient of an n-connected, n-dimensional simplicial complex. In the first case the group theoretic point of view leads to a Hodge type decomposition of the cohomology and we study this decomposition for cohomology classes that are Poincare duals of totally geodesic submanifolds. We use the second context to generalize to higher dimensions Margulis' construction of expander graphs as finite quotients of Cayley graphs of groups with Property (T).