Lattices with and lattices without spectral gap
Bachir Bekka, Alexander Lubotzky · Groups Geometry and Dynamics · 2011
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 G} to the orthogonal of the constants in L^2(\Gamma\backslash G) has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups G and lattices \Gamma for which L^2(\Gamma\backslash G) has no spectral gap. This answers in the negative a question asked by Margulis. In fact, G can be taken to be the group of orientation preserving automorphisms of a k -regular tree for k>2 .