Abstract: Classical Hodge theory says that the zero eigenspace of the Laplacian of a compact Riemannian manifold M gives us the de Rham cohomology. When M = Γ\G/K is a compact locally symmetric space, the Laplacian can be identified with the Casimir element in the universal enveloping algebra of the Lie algebra of G. This allows representation theoretic techniques for studying the cohomology of Γ\G/K. After stating our results in this context, we will move onto an analogous scenario in which manifolds are replaced by finite simplicial complexes X. When X is obtained as a quotient of a skeleton of a K(Γ, 1) space, where Γ is countable group, the Laplacian can again be identified with an element of the group algebra. In dimension 1 this allows translation of Property (T) of Γ to a gap in the spectrum of the graph Laplacian, giving Margulis’ celebrated construction of expander graphs. We will state our result that generalizes this construction to higher dimensions. Finally we will discuss how relative Property (T) of a pair of groups can be expressed in terms of Laplacians in the group algebra. This is an ongoing work attempting a generalization of Ozawa’s result, that gave a computable criterion to check Property (T) and was used to prove Property (T) of automorphism groups of free groups.

Meeting ID: 948 0044 7244
Passcode: 597063