The topology of moduli spaces of free group representations

TL;DR

This paper studies the topological structure of moduli spaces of free group representations, showing deformation retractions to compact groups and identifying specific homotopy types of character varieties.

Contribution

It demonstrates that certain complex character varieties strongly deformation retract to compact subgroup spaces and determines their homotopy types as spheres.

Findings

SL(3,C)-character variety of rank 2 free group is homotopic to an 8-sphere

SL(2,C)-character variety of rank 3 free group is homotopic to a 6-sphere

The G-character variety deformation retracts to the K-character space

Abstract

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3,C)-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character variety of a rank 3 free group is homotopic to a 6 sphere.