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 of G on L²(G/H) has a spectral gap, that is, the restriction of to the orthogonal complement of the constants in L²(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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-1-9,
author = {Bachir Bekka and Yves Cornulier},
title = {A spectral gap property for subgroups of finite covolume in Lie groups},
journal = {Colloquium Mathematicae},
volume = {120},
year = {2010},
pages = {175-182},
zbl = {1188.22006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-1-9}
}
Bachir Bekka; Yves Cornulier. A spectral gap property for subgroups of finite covolume in Lie groups. Colloquium Mathematicae, Tome 120 (2010) pp. 175-182. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-1-9/