On uniform lattices in real semisimple groups (Q2802131)

Let \(G\) be a semisimple Lie group. In this paper, the authors show that if two arithmetic lattices \(\Gamma,\Sigma\subset G\) are \(G\)-isospectral, i.e., the \(G\)-representations on \(L^2(\Gamma\backslash G)\) and \(L^2(\Sigma\backslash G)\) are isomorphic, and if \(\Gamma\) is cocompact in \(G\), then so is \(\Sigma\). The proof rests on the known fact that there is a continuous spectrum given by Eisenstein series, if \(\Sigma\backslash G\) is not compact. The assertion can easily be generalized to arithmetic lattices in different groups \(\Gamma\subset G\) and \(\Sigma\subset H\), when isospectrality is replaced by the weaker question of compactness, where a representation \(\pi\) of \(G\) is compact if \(\pi(f)\) is a compact operator whenever \(f\in C_c^\infty(G)\).NEWLINENEWLINEThis latter formulation can be generalized to arbitrary locally compact groups to become the question: Let \(\Gamma\) be a lattice in a locally compact group \(G\). Suppose that the right translation representation on \(L^2(\Gamma\backslash G)\) is compact. Does it follow that \(\Gamma\backslash G\) is compact?