# Volume of representations and mapping degree

## TL;DR

This paper studies the volume associated with representations of fundamental groups into Lie groups, showing finiteness under certain conditions and exploring applications related to mapping degrees and model geometries.

## Contribution

It introduces a new volume invariant for representations into Lie groups and proves its finiteness for manifolds when the group contains a cocompact semisimple subgroup.

## Key findings

- The set of volumes is finite for any fixed manifold under specified conditions.
- Examples from model geometries are analyzed.
- Applications involving mapping degrees are discussed.

## Abstract

Given a connected real Lie group and a contractible homogeneous proper $G$--space $X$ furnished with a $G$--invariant volume form, a real valued volume can be assigned to any representation $\rho\colon \pi_1(M)\to G$ for any oriented closed smooth manifold $M$ of the same dimension as $X$. Suppose that $G$ contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given $M$. From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed.