Volume invariant and maximal representations of discrete subgroups of Lie groups

TL;DR

This paper introduces a volume invariant for representations of lattices in semisimple Lie groups, generalizing previous concepts, and characterizes discrete, faithful representations through the maximality of this invariant, with specific exceptions.

Contribution

The paper defines a new volume invariant for lattice representations in semisimple Lie groups and proves its maximality characterizes discrete, faithful representations.

Findings

Maximal volume invariant corresponds to discrete, faithful representations.

The invariant generalizes Goldman’s volume invariant for uniform lattices.

Characterization holds except for nonuniform lattices in PSL(2,C).

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 $\Gamma\subset \mathrm{PSL_2 \mathbb{C}}$ a nonuniform lattice.