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

Abstract. Let G be the fundamental group of the complement of the torus knot of type (m, n). This has a presentation G = x, y | x m = y n . We find the geometric description of the character variety X(G) of characters of representations of G into SL(3, C), GL(3, C) and PGL(3, C). IntroductionSince the foundational work of Culler and Shalen [1], the varieties of SL(2, C)-characters have been extensively studied. Given a manifold M , the variety of representations of π 1 (M ) into SL(2, C) and the variety of characters of such representations both contain information of the topology of M . This is specially interesting for 3-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related. This can be used to study knots K ⊂ S 3 , by analysing the SL(2, C)-character variety of the fundamental group of the knot complement S 3 − K (these are called knot groups).For a very different reason, the case of fundamental groups of surfaces has also been extensively analysed [2,5,9], in this situation focusing more on geometrical properties of the moduli space in itself (cf. non-abelian Hodge theory).However, much less is known of the character varieties for other groups, notably for SL(r, C) with r ≥ 3. The character varieties for SL(3, C) for free groups have been described in [7,8]. In the case of 3-manifolds, little has been done. In this paper, we study the case of the torus knots K m,n of any type (m, n), which are the first family of knots where the computations are rather feasible. The case of SL(2, C)-character varieties of torus knots was carried out in [11,12]. For SL(3, C), the torus knot K 2,3 has been done in [3].In the case of SL(2, C)-character varieties of torus knot groups, only one-dimensional irreducible components appear. However, when we move to SL(3, C), we see components of different dimensions. In the case of torus knots, we shall see components of dimension 4 and of dimension 2. Our main result is an explicit geometrical description of the SL(3, C)-character variety of torus knots. Theorem 1.1. Let m, n be coprime positive integers. By swapping them if necessary, assume that n is odd. The SL(3, C)-character variety X 3 of the torus knot K m,n ⊂ S 3 is stratified into the following components:• One component consisting of totally reducible representations, isomorphic to C 2 . • If n is even, there are (m−1)/2 extra components consisting of partially reducible representations, each isomorphic to {(u, v) ∈ C 2 |v = 0, v = u 2 }. • 1 12 (n−1)(n−2)(m−1)(m−2) componens of dimension 4, consisting of irreducible representations, all isomorphic to each other, and which are described explicitly in Remark 8.5.• 1 2 (n − 1)(m − 1)(n + m − 4) components consisting of irreducible representations, each isomorphic to (C * ) 2 − {x + y = 1}.Moreover m, n can be recovered from the above information.We also describe geometrically how these components fit into the whole of the SL(3, C)-character variety, that is what is the closure of each of the strata in X 3 . In ...

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“…The character varieties for SL(3, C) for free groups have been described in [7,8]. In the case of 3-manifolds, little has been done.…”

Section: Introductionmentioning

confidence: 99%

“…is an isomorphism (see Chapter 1 in [8]). This is the same as to say that B = A G , that is, the ring of invariant polynomials is generated by characters.…”

Section: Introductionmentioning

confidence: 99%

Geometry of the SL(3, ℂ)–character variety of torus knots

Muñoz

Porti

2016

Algebr. Geom. Topol.

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“…Their main results were explicit formulas for the E-polynomials of character varieties of punctured Riemann surfaces of genus 1, 2 where the monodromy around the puncture is arbitrary, not necessarily semisimple. The techniques were used in [18] for two punctures and in [2] [16] for free groups, and they were later developed in [19] to deal with fibrations over bases of complex dimension two, where the case g = 3 was solved. Finally, in [20] the arbitrary genus polynomials were obtained inductively and formulas generalizing the results of [17], [19] were given.…”

Section: Introductionmentioning

confidence: 99%

E-Polynomials of the SL(2, ℂ)-Character Varieties of Surface Groups

Martínez

Muñoz

2015

Int Math Res Notices

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“…Remark 2. In [12] the authors gave a computation of the SL 3 (C)-character variety of free groups. We observe that although this paper uses a different method of computing E-polynomials, there are clearly some similarities between their computations and the computations given in this section.…”

Section: Sl 3 Gl 3 Character Varieties For Fundamental Groups Of Comentioning

confidence: 99%

“…The E-polynomial for the SL 2 (C) case was computed in [1] and the GL 2 (C) case in [20]. We note that our method can also be used to compute the E-polynomials in the SL 3 (C) and GL 3 (C) cases, which were computed in [12] and [20], respectively. Our method recovers these results.…”

Section: Introductionmentioning

confidence: 99%

Arithmetic of singular character varieties and their E‐polynomials

Baraglia

Hekmati

2017

Proc. London Math. Soc.

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We calculate the E-polynomials of the SL3(C)and GL3(C)-character varieties of compact oriented surfaces of any genus and the E-polynomials of the SL2(C)and GL2(C)-character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the E-polynomials of the SL2(C)-character varieties of compact orientable surfaces, which were computed by Logares, Muñoz and Newstead for genus g = 1, 2 and by Martinez and Muñoz for g 3. Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.

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

Cited by 19 publications

References 28 publications

Geometry of the SL(3, ℂ)–character variety of torus knots

Geometry of the SL(3, ℂ)–character variety of torus knots

E-Polynomials of the SL(2, ℂ)-Character Varieties of Surface Groups

Arithmetic of singular character varieties and their E‐polynomials

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