Vol. 282, No. 1, 2016

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$E$-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
Abstract

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3, ). 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
Authors
Sean Lawton
Department of Mathematical Sciences
George Mason University
4400 University Drive
Fairfax, VA 22030
United States
Vicente Muñoz
Facultad de Matemáticas
Universidad Complutense de Madrid
Plaza de Ciencias 3
28040 Madrid
Spain