Lattices with and lattices without spectral gap
Lattices with and lattices without spectral gap
Let G=\boldsymbol{G}(\mathbb{k}) be the \mathbb{k} -rational points of a simple algebraic group \boldsymbol{G} over a local field \mathbb{k} and let \Gamma be a lattice in G . We show that the regular representation \rho_{\Gamma\backslash G} of G on L^2(\Gamma\backslash G) has a spectral gap, that is, the restriction of \rho_{\Gamma\backslash …