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

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