$E$-polynomial of the $\mathrm{SL}(3,\mathbb{C})$-character variety of free groups

Sean Lawton and Vicente Muñoz

Vol. 282 (2016), No. 1, 173–202

DOI: 10.2140/pjm.2016.282.173

Abstract

We compute the $E$-polynomial of the character variety of representations of a rank $r$ free group in $SL\left(3,ℂ\right)$. Expanding upon techniques of Logares, Muñoz and Newstead (Rev. Mat. Complut. 26:2 (2013), 635–703), 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 a new proof of results of Cavazos and Lawton (Int. J. Math. 25:6 (2014), 1450058).

Keywords

$E$-polynomial, free group, $\mathrm{SL}(3,\mathbb{C})$, character variety

Mathematical Subject Classification 2010

Primary: 14D20, 20C15, 14L30, 20E05

Milestones

Received: 24 March 2015

Revised: 29 July 2015

Accepted: 8 September 2015

Published: 24 February 2016