Varieties of Characters

Sean Lawton George Mason University Fall Eastern Sectional Meeting September 25, 2016 Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Any and every (affine) algebraic C-group G that arises in this fashion is called reductive. Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Any and every (affine) algebraic C-group G that arises in this fashion is called reductive. We will say a Zariski dense subgroup G G is real reductive if G(R) 0 G G(R). Lawton (GMU) (AMS, September 2016)

Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Any and every (affine) algebraic C-group G that arises in this fashion is called reductive. We will say a Zariski dense subgroup G G is real reductive if G(R) 0 G G(R). Ex: SL(n, C) or SL(n, R) or SU(n) or SO(n) Lawton (GMU) (AMS, September 2016)

Step 2: Homomorphisms Let R Γ (G) = Hom(Γ, G), and denote r {}}{ G G G by G r. Lawton (GMU) (AMS, September 2016)

Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Lawton (GMU) (AMS, September 2016)

Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). Lawton (GMU) (AMS, September 2016)

Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). It is not generally surjective since not all r-tuples of elements of G will satisfy r i (g 1,..., g r ) = 1 for all i I. Lawton (GMU) (AMS, September 2016)

Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). It is not generally surjective since not all r-tuples of elements of G will satisfy r i (g 1,..., g r ) = 1 for all i I. One notable exception is the case when there are no relations r i, that is, when Γ is a free group. Lawton (GMU) (AMS, September 2016)

Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). It is not generally surjective since not all r-tuples of elements of G will satisfy r i (g 1,..., g r ) = 1 for all i I. One notable exception is the case when there are no relations r i, that is, when Γ is a free group. However, this does show that Hom(Γ, G) = {(g 1,..., g r ) G r r i (g 1,..., g r ) = 1, i I } G r is a subvariety of the smooth variety G r. Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. If G is real algebraic, then the character variety is semi-algebraic. Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. If G is real algebraic, then the character variety is semi-algebraic. If G is compact, then the character variety is compact and is just the usual orbit space of Hom(Γ, G). Lawton (GMU) (AMS, September 2016)

Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. If G is real algebraic, then the character variety is semi-algebraic. If G is compact, then the character variety is compact and is just the usual orbit space of Hom(Γ, G). If G is complex reductive, then the character variety is identified with the GIT quotient. Lawton (GMU) (AMS, September 2016)

Why care? If you Google any of the following key words, you will find that the study of character varieties at least touches the following topics: flat G-bundles, G-Higgs bundles, holomorphic vector bundles, (G, X )-structures, Mirror symmetry, String vacua, Yang-Mills connections, Convex Cocompactness, knot invariants, Geometric Langlands, Quantization, Spin Networks, A-polynomial, hyperbolic manifolds Lawton (GMU) (AMS, September 2016)

Examples Figure : X Z (SL(2, R)) 2 1 0 1 2 1 0 1 2 3 Figure : X Z (SU(3)) Lawton (GMU) (AMS, September 2016)

Figure : X Z Z (SU(2)) Figure : X Z Z (SL(2, C))(R) with boundary value 2. Lawton (GMU) (AMS, September 2016)

X Z 3(SU(2)) is a 3 dimensional orbifold with 8 singularities; each locally C R (RP 2 ). Lawton (GMU) (AMS, September 2016)

History The term character variety arose from the seminal work of Peter Shalen and Marc Culler in Varieties of group representations and splittings of 3-manifolds, published in the Annals of Mathematics in 1983. Lawton (GMU) (AMS, September 2016)

History The term character variety arose from the seminal work of Peter Shalen and Marc Culler in Varieties of group representations and splittings of 3-manifolds, published in the Annals of Mathematics in 1983. One of the theorems they showed in this paper is that the space of SL(2, C)-characters was a variety. Lawton (GMU) (AMS, September 2016)

History The term character variety arose from the seminal work of Peter Shalen and Marc Culler in Varieties of group representations and splittings of 3-manifolds, published in the Annals of Mathematics in 1983. One of the theorems they showed in this paper is that the space of SL(2, C)-characters was a variety...hence the term. Lawton (GMU) (AMS, September 2016)

Variety of Characters Fix a connected reductive subgroup G of SL(n, C). Lawton (GMU) (AMS, September 2016)

Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). Lawton (GMU) (AMS, September 2016)

Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). Lawton (GMU) (AMS, September 2016)

Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). By thinking of SO(2, C), one can deduce that character varieties are not generally varieties of characters, Lawton (GMU) (AMS, September 2016)

Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). By thinking of SO(2, C), one can deduce that character varieties are not generally varieties of characters, and by work of Sikora this generalizes to SO(2n, C). Lawton (GMU) (AMS, September 2016)

Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). By thinking of SO(2, C), one can deduce that character varieties are not generally varieties of characters, and by work of Sikora this generalizes to SO(2n, C). For example, X F2 (SO(2, C)) is homotopic to S 1 S 1 while Ch F2 (SO(2, C)) is homotopic to S 2. Lawton (GMU) (AMS, September 2016)

Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Lawton (GMU) (AMS, September 2016)

Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Therefore, C[X Γ (SL(n, C))] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic unimodular matrices. Lawton (GMU) (AMS, September 2016)

Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Therefore, C[X Γ (SL(n, C))] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic unimodular matrices. More generally, T Γ (G) coincides with C[X Γ (G)] for any Γ and G equal to SL(m, C), Sp(2m, C), or SO(2m + 1, C); a result of Lawton and also Sikora. Lawton (GMU) (AMS, September 2016)

Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Therefore, C[X Γ (SL(n, C))] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic unimodular matrices. More generally, T Γ (G) coincides with C[X Γ (G)] for any Γ and G equal to SL(m, C), Sp(2m, C), or SO(2m + 1, C); a result of Lawton and also Sikora. How do they relate in general? Lawton (GMU) (AMS, September 2016)

The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Lawton (GMU) (AMS, September 2016)

The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). Lawton (GMU) (AMS, September 2016)

The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). π : N (G) Aut T (G) given by h C h where C h (g) = hgh 1. Lawton (GMU) (AMS, September 2016)

The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). π : N (G) Aut T (G) given by h C h where C h (g) = hgh 1. Since Ker(π) contains the center of SL(n, C), π is not one-to-one. Lawton (GMU) (AMS, September 2016)

The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). π : N (G) Aut T (G) given by h C h where C h (g) = hgh 1. Since Ker(π) contains the center of SL(n, C), π is not one-to-one. However, π is onto and Out T (G) is finite. Lawton (GMU) (AMS, September 2016)

Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Lawton (GMU) (AMS, September 2016)

Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Lawton (GMU) (AMS, September 2016)

Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Lawton (GMU) (AMS, September 2016)

Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Since there is a unique closed orbit in each equivalence class in X F2 (SL(n, C)), ρ and αρ are conjugate in SL(n, C). Lawton (GMU) (AMS, September 2016)

Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Since there is a unique closed orbit in each equivalence class in X F2 (SL(n, C)), ρ and αρ are conjugate in SL(n, C). Thus, since ρ is irreducible, this implies that α : G G coincides with conjugation of G by some element of SL(n, C). Lawton (GMU) (AMS, September 2016)

Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Since there is a unique closed orbit in each equivalence class in X F2 (SL(n, C)), ρ and αρ are conjugate in SL(n, C). Thus, since ρ is irreducible, this implies that α : G G coincides with conjugation of G by some element of SL(n, C). Now, Out T (G) is the epimorphic image of N (G)/G which is finite by a result of Vinberg. Lawton (GMU) (AMS, September 2016)

Theorem (Lawton-Sikora, 2016) Let G SL(n, C) be a connected reductive group, and Γ a finitely generated group. Then: 1 Out T (G) acts freely on X i Γ (G). 2 ϕ i : X i Γ (G)/Out T (G) ChΓ i (G) is a finite, birational bijection. 3 If X i Γ (G)/Out T (G) is normal, then ϕ i is a normalization map. Lawton (GMU) (AMS, September 2016)

Theorem (Lawton-Sikora, 2016) Let G SL(n, C) be a connected reductive group, and Γ a finitely generated group. Then: 1 Out T (G) acts freely on X i Γ (G). 2 ϕ i : X i Γ (G)/Out T (G) ChΓ i (G) is a finite, birational bijection. 3 If X i Γ (G)/Out T (G) is normal, then ϕ i is a normalization map. So although character varieties are not varieties of characters in general, we do have: [X i Γ (G)/Out T (G)] = [Ch i Γ (G)]. Lawton (GMU) (AMS, September 2016)

Corollaries Corollary Under the assumptions of the last Theorem, we have: 1 If X Γ (G)/Out T (G) is irreducible and contains an irreducible representation, then ϕ is a birational morphism. 2 If X Γ (G)/Out T (G) is normal and contains an irreducible representation, then ϕ is a normalization map. 3 ϕ is a normalization map for Γ a free group of rank r 2. 4 ϕ is a normalization map for Γ a genus g 2 surface group and G = SL(m, C) or GL(m, C), m 1. 5 ϕ is a normalization map for Γ a free abelian group of rank r 1 and G = SL(m, C) or Sp(2m, C), m 1. Lawton (GMU) (AMS, September 2016)

Corollary Let F r be a free group of rank r 2. If G is a connected reductive subgroup of SL(n, C) such that all simple factors of the Lie algebra of the derived subgroup [G, G] have rank 2 or more, then X sm F r (G) is étale equivalent to ChF sm r (G). Lawton (GMU) (AMS, September 2016)

Thank you! References are at http://arxiv.org/a/lawton_s_1. I gratefully acknowledge support from: Lawton (GMU) (AMS, September 2016)