# The topology of Baumslag-Solitar representations

## TL;DR

This paper demonstrates a strong deformation retraction of the representation space of certain Baumslag-Solitar groups into complex reductive groups onto the space into their maximal compact subgroups, revealing topological structure.

## Contribution

It establishes a deformation retraction for representations of Baumslag-Solitar groups into complex reductive groups under specific conditions on p and q.

## Key findings

- Deformation retraction exists when p and q are coprime with distinct absolute values.
- The representation space deformation retracts onto the compact subgroup representations.
- Provides insight into the topology of representation varieties for Baumslag-Solitar groups.

## Abstract

Let $\Gamma=\langle a,b | a b^{p} a^{-1} = b^{q}\rangle$ be a Baumslag--Solitar group and $G$ be a complex reductive algebraic group with maximal compact subgroup $K<G$. We show that, when $p$ and $q$ are relatively prime with distinct absolute values, there is a strong deformation retraction retraction of $Hom(\Gamma,G)$ onto $Hom(\Gamma,K)$.