E-polynomial of SL(2,C)-Character Varieties of Free groups

Abstract

Let ${\mathsf{F}}_r$ be a free group of rank $r$, ${\mathbb{F}}_q$ a finite field of order q, and let ${\mathrm{SL}}_n({\mathbb{F}}_q)$ act on $\mathrm{Hom}({\mathsf{F}}_r,{\mathrm{SL}}_n({\mathbb{F}}_q))$ by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits $\mathrm{Hom}({\mathsf{F}}_r,{\mathrm{SL}}_n({\mathbb{F}}_q))/{\mathrm{SL}}_n({\mathbb{F}}_q)$. Our first main theorem is the implementation of this algorithm in the case $n=2$. As an application, we determine the $E$-polynomial of the character variety $\mathrm{Hom}({\mathsf{F}}_r,{\mathrm{SL}}_2(\mathbb{C}))//!/{\mathrm{SL}}_2(\mathbb{C})$, and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.

23 pages; v2 has a more streamlined exposition, and includes additional references; to appear in International Journal of Mathematics

References in corpus (1)

• The topology of moduli spaces of free group representations

Cited by in corpus (6)

• Arithmetic of singular character varieties and their $E$-polynomials
• E-polynomial of the SL(3,C)-character variety of free groups
• Serre polynomials of $S{L}_n$- and $PG{L}_n$-character varieties of free groups
• The moduli of representations of degree 2
• Representation varieties of twisted Hopf links
• Pseudo-quotients of algebraic actions and their application to character varieties