3:00 pm Friday, February 20, 2009
Special Algebra+Geometry+Topology Seminar: Length commensurable and isospectral locally symmetric spaces by Gopal Prasad (University of Michigan) in RLM 10.176
In this talk I will describe some recent joint work with Andrei Rapinchuk on weakly commensurable arithmetic groups in semi- simple groups, and describe applications of our results to a study of locally symmetric spaces. We show that for the symmetric space X of a noncompact absolutely simple real Lie group G, if the quotients X/\Gamma_1 and X/\Gamma_2 are length-commensurable or compact and isospectral, where \Gamma_1 and \Gamma_2 are lattices in G, and at least one of them is arithmetic, then the subgroups \Gamma_1 and \Gamma_2 are weakly commensurable. Our results on weakly commensurable arithmetic groups then imply that length-commensurability or isospectrality of X/\Gamma_1 and X/\Gamma_2 implies that both \Gamma_1 and \Gamma_2 are arithmetic, and if G is of type other than A_n, D_{2n+1}, D_4 and E_6, then these manifolds are commensurable, that is they admit a common finite cover. Submitted by
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