On the equivalence of the definitions of volume of representations

Sungwoon Kim

Vol. 280 (2016), No. 1, 51–68

DOI: 10.2140/pjm.2016.280.51

Abstract

Let $G$ be a rank-$1$ simple Lie group and let $M$ be a connected, orientable, aspherical, tame manifold. Assume that each end of $M$ has amenable fundamental group. There are several definitions of volume of representations of ${\pi }_1\left(M\right)$ into $G$. We give a new definition of volume of representations and furthermore show that all definitions so far are equivalent.

Keywords

Lie groups, representation, volume, symmetric spaces, disjoint cone

Mathematical Subject Classification 2010

Primary: 53C24, 53C35

Milestones

Received: 6 January 2015

Accepted: 30 April 2015

Published: 28 December 2015