The Topology of Parabolic Character Varieties of Free Groups

TL;DR

This paper investigates the topology of parabolic character varieties of free groups, proving that certain representation spaces deformation retract onto compact subspaces, thus revealing their homotopy equivalence and providing explicit examples.

Contribution

It establishes a strong deformation retraction from the complex representation space to a compact subgroup, extending understanding of parabolic character varieties.

Findings

Y is a strong deformation retraction of X

X and Y are homotopy equivalent

Explicit examples relate X to relative character varieties

Abstract

Let G be a complex affine algebraic reductive group, and let K be a maximal compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or equal to 0, let X (respectively, Y) be the space of equivalence classes of representations of the free group of m+n generators in G (respectively, K) such that for each i between 1 and m, the image of the i-th free generator is conjugate to h_i. These spaces are parabolic analogues of character varieties of free groups. We prove that Y is a strong deformation retraction of X. In particular, X and Y are homotopy equivalent. We also describe explicit examples relating X to relative character varieties.