Volume invariant and maximal representations of discrete subgroups of Lie groups

Abstract

Let $\Gamma$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma$ into $G$ except for $\Gamma \subset \mathrm{PS}{\mathrm{L}}_2\mathbb{C}$ a nonuniform lattice.

28 pages