E-polynomial of the SL(3,C)-character variety of free groups

TL;DR

This paper calculates the E-polynomial of the SL(3,C)-character variety for free groups, providing detailed stratifications and extending techniques to include smooth, singular, and abelian loci, with new proofs for related cases.

Contribution

It introduces a comprehensive method to compute E-polynomials of SL(3,C) character varieties, extending existing techniques and providing new proofs for related groups.

Findings

E-polynomial of the SL(3,C) character variety computed

Stratification of the representation space achieved

E-polynomials of various loci determined

Abstract

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3, C). Expanding upon existing techniques, we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give new proofs of the SL(2,C), PGL(2,C), and PGL(3,C) cases.