A spectral gap property for subgroups of finite covolume in Lie groups

Details A spectral gap property for subgroups of finite covolume in Lie groups A spectral gap property for subgroups of finite covolume in Lie groups

2009-03-25

arxiv.org/abs/0903.4367v1

Colloq. Math. 118 (2010), 175-182

Mathematics

Group Theory

9 pages, no figure

Scientific paper

10.4064/cm118-1-9

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation lambda_{G/H} of G on L^2(G/H) has a spectral gap, that is, the restriction of lambda_{G/H} to the orthogonal of the constants in L^2(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

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