Heilbronn Algebra Seminar - Henrique Mendes da Silva e Souza

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Title: Cohomological asymptotics for linear groups Abstract: For a finitely generated subgroup H of a complex semisimple Lie group G, we investigate how the cohomology groups H^i(H,V) grow as V ranges over thefinite-dimensional complex representations of G. Assuming a mild finiteness condition on H, we formulate a conjecture relating the dimensions of H^i(H,V) to the L²-Betti numbers of H. We then explain how this conjecture is connected to the p-adic representation theory of G?, the principal congruence subgroup of the simply connected p-adic split semisimple algebraic group having the same root system as G. We'll finish with a discussion of known cases (joint with L. Guerrero) as well as recent developments.