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In the last section we compute the the trace-preserving outer automor- phisms of G for various simple complex Lie groups.

Acknowledgements. We acknowledge support from U.S. National Science Foundation grants: DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network) and Grant No. 0932078000 while we were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2015 semester. Lawton was also partially supported by the grant from the Simons Foun- dation (Collaboration #245642) and the U.S. National Science Foundation (DMS #1309376). Lastly, we thank the referee for helping improve the paper.

2. Variety of Characters Let Γ be a finitely generated group. Fix a connected reductive sub- group G of SL(n, C). A G-character of Γ, denoted χρ, is the trace of a G-representation of Γ: ρ  Γ / G / SL(n, C) tr / C . Denote the set of G-characters of Γ by Ch(Γ, G). Let Hom(Γ, G) and X (Γ, G) = Hom(Γ, G)//G be the representation vari- ety and the G-, respectively (considered as affine varieties). Let T (Γ, G) be the trace G-algebra of Γ, that is, the subalgebra of C[X (Γ, G)] generated by trace functions τγ for γ ∈ Γ defined by τγ(ρ) = tr(ρ(γ)), see [CS83, FL11, Sik13] for more details. It was proven in [FL11, Appendix A] and in [Sik13, Thm. 5] that 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) . For the reader’s convenience, we include a concise proof of that result here. Proposition 1. For G equal to SL(m, C), Sp(2m, C), or SO(2m + 1, C), and for all finitely generated Γ, T (Γ, G)= C[X (Γ, G)].

Proof. If Γ is a group on r generators, then the Fr → Γ mapping the free generators of Fr to the generators of Γ yields an em- bedding X (Γ, G) → X (Fr, G) and the corresponding dual epimorphism C[X (Fr, G)] → C[X (Γ, G)]. Since that map sends trace functions to trace r G functions, it is enough to show that C[X (Fr, G)] = C[G ] is generated by trace functions. By the assumptions of the theorem, G is an algebraic of SL(n, C) for some n and, hence, of the space of the n×n complex matrices Mat(n, C). The group G acts on it by conjugation and the embedding G ⊂ SL(n, C) ⊂ Mat(n, C), induces a G-equivariant epimorphism C[Mat(n, C)r] → C[Gr], which restricts to an epimorphism C[Mat(n, C)r]G → C[Gr]G by the exis- tence of Reynolds operators. Now the statement follows from the results VARIETIESOFCHARACTERS 3 of [Pro76] stating that C[Mat(n, C)r]G is generated by traces of monomi- als in matrices and their inverses for G = SL(m, C), SO(2m + 1, C), and Sp(2m, C). 

The above statement does not hold for G = SO(2m, C), m ≥ 1. In that case C[X (Γ, G)] is a T (Γ, G)-algebra finitely generated by expressions in- volving Pfaffians, [ATZ95], [Sik15, Proposition 16]. In this paper we intend to elaborate on the relationship between coordi- nate rings of character varieties and their trace algebras. Denote by ϕ the natural projection map ϕ : Hom(Γ, G)//N (G) →X (Γ, SL(n, C)), where N (G) is the normalizer of G in SL(n, C) . Since a normalizer of a con- nected is reductive, the above GIT quotient is well defined. By [Vin96, Corollary 2 to Theorem 1], ϕ is a finite morphism and, hence, its is closed. Lemma 2. There is an

ψ : T (Γ, G) → C[ϕ(Hom(Γ, G)//N (G))] sending regular functions τγ to regular functions τ˜γ on ϕ(Hom(Γ, G)//N (G)) defined by τ˜γ ([ρ]) = tr(ρ(γ)). Since τγ generate T (Γ, G), this condition de- fines ψ uniquely.

Proof. Since any polynomial identity in τγ’s on X (Γ, G) holds if and only if it holds on ϕ(Hom(Γ, G)//N (G)) as well, the above homomorphism is well defined and injective. Since C[X (Γ, SL(n, C))] is generated by trace functions, C[ϕ(Hom(Γ, G)//N (G))] is generated by trace functions as well, and hence, ψ is onto. 

We have a map Ψ : Ch(Γ, G) → Spec T (Γ, G) sending χρ to a algebra homomorphism T (Γ, G) → C, determined by τγ 7→ χρ(γ). It is not hard to see that Ψ is well defined. Furthermore, we have: Proposition 3. Ψ is a bijection.

Proof. If Ψ(χρ) = Ψ(χρ′ ) then χρ(γ) = χρ′ (γ) for all γ ∈ Γ, implying χρ = χρ′ . Thus, Ψ is injective. By Lemma 2, it remains to prove that Ψ : Ch(Γ, G) → ϕ(Hom(Γ, G)//N (G)) is surjective. Indeed, for every ρ ∈ Hom(Γ, G), Ψ(χρ)= ϕ(ρ) and, hence, Ψ is onto.  4 S. LAWTON AND ADAM S. SIKORA

The above proposition provides for the structure of an algebraic set on Ch(Γ, G) with its coordinate ring being T (Γ, G). By analogy with the G- character variety of Γ, we will call Ch(Γ, G) the variety of G-characters of Γ or simply the variety of characters when the context is clear. By Lemma 2, ϕ(Hom(Γ, G)//N (G)) = Ch(Γ, G) and, hence, ϕ can be written as ϕ : Hom(Γ, G)//N (G) →Ch(Γ, G). Following [AB94], denote the set of trace-preserving automorphisms of G by AutT (G), that is, the automorphisms α ∈ Aut(G) such that for all g ∈ G, tr(α(g)) = tr(g). Note that AutT (G) acts naturally on Hom(Γ, G) by (α, ρ) 7→ α◦ρ and that this action descends to an action of AutT (G)/Inn(G) on X (Γ, G), where Inn(G) is the inner automorphism group of G. We have a natural map

π : N (G) → AutT (G) −1 given by h 7→ Ch where Ch(g) = hgh . Since Ker(π) contains the of SL(n, C), π is not one-to-one. Recall that ρ :Γ → G is irreducible if its image is not properly contained in any parabolic subgroup of G, see for example [Sik12]. Lemma 4. π is onto.

Proof. Let Gc be a maximal compact subgroup of G. By [Gel08, Lemma 1.8], there exists a representation ρ : F2 → Gc with a dense image. Then the -image of ρ in G is Zariski dense and, in particular, ρ : F2 → Gc ֒→ G is irre ducible. Let α ∈ AutT (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(F2, SL(n, C)), see [LM85, Theorem 1.27]. Since there is a unique closed orbit in each equivalence class in X (F2, SL(n, C)), see for example [Dol03, Corollary 6.1], ρ and αρ are conjugate in SL(n, C). Thus, since ρ has Zariski dense image in G, this implies that α : G → G coincides with conjugation of G by some element of SL(n, C). 

We denote

OutT (G) := AutT (G)/Inn(G), and call this group the trace-preserving outer automorphisms.

Lemma 5. OutT (G) is finite.

Proof. OutT (G) is a subgroup of Out(G) and, consequently, of Out(g) which coincides with the automorphism group of the Dynkin diagram of g. Conse- quently, OutT (G) is finite for semisimple G. For the general not semisimple case, note that OutT (G) is the epimorphic image of N (G)/G which is finite by [Vin96, Corollary 3b].  VARIETIESOFCHARACTERS 5

Clearly OutT (G) descends to an action on X (Γ, G) yielding an isomor- phism Hom(Γ, G)//N (G) →X (Γ, G)/OutT (G). Consequently, ϕ becomes a map

(1) ϕ : X (Γ, G)/OutT (G) →Ch(Γ, G). By [Vin96, Corollary 2 of Theorem 1], the map X (Γ, G) → Ch(Γ, G) is finite. Consequently, ϕ is finite too. Denote the set of G-representations of Γ having Zariski dense image by Homzd(Γ, G), and let X zd(Γ, G) be the corresponding subspace in X (Γ, G). Let Chzd(Γ, G) ⊂Ch(Γ, G) be the of characters of Zariski dense repre- zd zd sentations of Γ, and let ϕ be the restriction of (1) to X (Γ, G)/OutT (G) ⊂ X (Γ, G)/OutT (G). The example following [AB94, Proposition 8.2] shows that if G is reduc- tive, then Homzd(Γ, G) may not be Zariski open. However, the following lemma is true. Lemma 6. Let G be connected and reductive. Consider the S: Homzd(Γ, G) ⊂ Hom(Γ, G), Chzd(Γ, G) ⊂ Ch(Γ, G), X zd(Γ, G) ⊂ X (Γ, G), zd and X (Γ, G)/OutT (G) ⊂X (Γ, G)/OutT (G). (a) If G is semisimple, then the subsets in S are Zariski open. (b) Homzd(Γ, G) 6= ∅, whenever Γ maps onto a free group of rank at least 2. (c) If Homzd(Γ, G) 6= ∅, then the subsets S are dense (and hence Zariski dense) whenever the corresponding superset in S is irreducible. Proof. (a) When G is semisimple Homzd(Γ, G) is Zariski open by [AB94, Proposition 8.2]. Let πG : Hom(Γ, G) →X (Γ, G) be the GIT quotient map. Because X (Γ, G) has the quotient topology from πG : Hom(Γ, G) →X (Γ, G) for the openness of X zd(Γ, G) is enough to show that −1 zd zd (2) πG (X (Γ, G)) = Hom (Γ, G). The inclusion ⊃ is obvious. To show ⊂, note that every Zariski dense ρ is irreducible and, hence, a stable point of the action of G by conjugation, by [Sik12, Corollary 31]. By [Dol03] (Theorem 8.1 and the Property (iv) of a −1 good categorical quotient), πG ([ρ]) is the (set-theoretic) G-orbit of ρ. Now the inclusion ⊂ of (2) follows from the fact that all representations in that orbit (that is, conjugates of ρ) are Zariski dense. zd The proof of openness of X (Γ, G)/OutT (G) is very similar to the above one, but simpler. Again, considering the quotient map

πOutT (G) : X (Γ, G) →X (Γ, G)/OutT (G), it is enough to show that −1 zd zd πOutT (G)(X (Γ, G)/OutT (G)) = X (Γ, G), 6 S. LAWTON AND ADAM S. SIKORA but since OutT (G) is finite the statement follows from the fact that this categorical quotient coincides with the set-theoretic quotient. We now show Chzd(Γ, G) is open. Let Y,Y zd and Y nzd respectively denote zd X (Γ, G)/OutT (G), X (Γ, G)/OutT (G) and the complement nzd zd X (Γ, G)/OutT (G) := X (Γ, G)/OutT (G) −X (Γ, G)/OutT (G). Since ϕ is onto, Ch(Γ, G)= ϕ(Y )= ϕ(Y zd) ∪ ϕ(Y nzd). We have shown that Y nzd is closed and, since ϕ is finite, ϕ(Y nzd) is closed in Ch(Γ, G). Now the openness of Chzd(Γ, G)= ϕ(Y zd) follows from [AB94, Proposition 8.1] which implies that ϕ(Y zd) and ϕ(Y nzd) are disjoint. (b) When Γ maps onto a free group of rank at least 2, then by [Gel08, Lemma 1.8] Homzd(Γ, G) is non-empty. (c) Since G is connected and reductive, G =∼ DG×F T where DG = [G, G] is semisimple, T is a central algebraic torus, and F = T ∩ DG is a finite central subgroup. Since Hom(Γ, G) is connected, Hom(Γ, G) =∼ Hom0(Γ, DG × T )/Hom(Γ, F ) ′ =∼ (Hom0(Γ, DG) × Hom0(Γ, T ))/Hom (Γ, F ), by [Sik15, Prop. 5(1)], where the superscript 0 denotes the connected com- ′ ponent of the trivial representation and Hom (Γ, F ) denotes the subgroup of 0 Hom(Γ, F ) mapping Hom (Γ, DG×T ) to itself. A subgroup of G =∼ DG×F T ′ ′ is Zariski dense, if and only if it is of the form D ×F T for Zariski dense D′ ⊂ DG, T ′ ⊂ T. Consequently, ′ Homzd(Γ, G) =∼ (Hom0,zd(Γ, DG) × Hom0,zd(Γ, T ))/Hom (Γ, F ). Since Homzd(Γ, G) 6= ∅, Hom0,zd(Γ, DG) is non-empty and also Zariski open in Hom0(Γ, DG) by Part (a). The irreducibility of Hom(Γ, G) implies the irreducibility of Hom0(Γ, DG) and, therefore, Hom0,zd(Γ, DG) is dense in Hom0(Γ, DG). Note that Hom0(Γ, T ) =∼ Hom0(Γ/[Γ, Γ], T ) =∼ T r, where r is the rank of the free part of the abelianization Γ/[Γ, Γ]. Since the set of elements of T r generating Zariski dense in T r is Zariski dense, Hom0,zd(Γ, T ) is Zariski dense in Hom0(Γ, T ). Thus, Homzd(Γ, G) is Zariski dense in Hom(Γ, G). Similarly, X zd(Γ, G) is dense in its superset, by the same argument cou- zd pled with [Sik15, Prop. 5(2)]. That implies X (Γ, G)/OutT (G) is dense zd as well. Finally, since ϕ is onto, it maps the dense set X (Γ, G)/OutT (G) onto a dense set Chzd(Γ, G).  Theorem 7. Let G ⊂ SL(n, C) be a semisimple group, and Γ a finitely generated group. Then: zd (1) OutT (G) acts freely on X (Γ, G). VARIETIESOFCHARACTERS 7

zd zd zd (2) ϕ : X (Γ, G)/OutT (G) → Ch (Γ, G) is a finite, birational bijec- tion. zd zd (3) If X (Γ, G)/OutT (G) is normal, then ϕ is a normalization map.

Proof. (1) Suppose for [α] ∈ OutT (G) that [α] · [ρ] = [ρ] for some [ρ] ∈ X zd(Γ, G). By the argument of the proof of Lemma 6(a), ρ and αρ being equivalent and Zariski dense, are conjugate. In other words, α(ρ(γ)) = hρ(γ)h−1 for some h ∈ G and all γ ∈ Γ. Since ρ has Zariski dense image, α(g)= hgh−1 for all g ∈ G. In particular, α ∈ Inn(G) and [α] is the identity in OutT (G). (2) Since ϕzd is a restriction of a finite map to a Zariski open set, it is also finite. It is injective by [AB94, Proposition 8.1], and ϕzd is onto by definition. Then, by [Vin96, Lemma 1], ϕzd is birational. Lastly, (3) is an immediate consequence of (2).  Example 8. Let D be the diagonal matrices in G = SO(2n, C). Then D forms an irreducible subgroup by [Sik12, Example 21]. Now take any A ∈ O(2n) with det(A) = −1. Then the conjugation by A is a non-trivial outer automorphism of SO(2n, C) which preserves D. Hence, OutT (G) does not act freely on irreducible representations. Thus, we cannot extend the previous theorem to the irreducible locus. It is a general fact that bijective morphisms f : X → Y give equality [X] = [Y ] in the Grothendieck ring K0(V ar/C), for example see [BBS13, Page 115]. Consequently, Theorem 7(2) implies the following corollary.

zd Corollary 9. Under the assumptions of Theorem 7, [X (Γ, G)/OutT (G)] = zd [Ch (Γ, G)] in the Grothendieck ring K0(V ar/C). Theorem 10. Let G ⊂ SL(n, C) be a connected reductive group, and Γ a finitely generated group. Then:

(1) If X (Γ, G)/OutT (G) is irreducible and contains a representation with Zariski dense image, then ϕ is a birational morphism. (2) If X (Γ, G)/OutT (G) is normal and contains a representation with Zariski dense image, then ϕ is a normalization map. (3) ϕ is a normalization map for Γ a free group of rank r ≥ 2 (compare [Vin96]). (4) ϕ is a normalization map for a genus g ≥ 2 surface group Γg and G = SL(m, C) or GL(m, C), m ≥ 1. (5) ϕ is a normalization map for Γ a free of rank r ≥ 1 and G = SL(m, C), or Sp(2m, C), m ≥ 1. Proof. Since we are assuming the set of representations with Zariski dense image is non-empty and X (Γ, G)/OutT (G) is irreducible as an algebraic set, zd X (Γ, G)/OutT (G) is dense in X (Γ, G)/OutT (G) by Lemma 6(c). Now (1) follows from [AB94, Proposition 8.1] and [Vin96, Lemma 1]. Then (2) is an immediate consequence of (1) and the finiteness of ϕ. 8 S. LAWTON AND ADAM S. SIKORA

Since Hom(Fr, G) is smooth, and X (Fr, G) and X (Fr, G)/OutT (G) are subsequent GIT quotients, all three spaces are irreducible and normal. Since there is a G-representation of Fr for any r ≥ 2 with Zariski dense image by Lemma 6(b), (3) follows from (2). By [Sim94a, Sim94b], X (Γg, G) is normal for surface groups Γg for g ≥ 2 when G is SL(n, C) or GL(n, C). The epimorphism Γg → Fg together with Lemma 6 yields a Zariski dense representation of Γg → G for every g ≥ 2. Now (4) follows from (2). Lastly, by [Sik14] X (Zr, G) is irreducible and normal for G equal to SL(m, C), or Sp(2m, C). In these cases OutT (G) is trivial (see Section 3), and by Proposition 1 X (Zr, G)= Ch(Zr, G). Hence, (5) follows.  In particular, for Γ and G from (1) in Corollary 10, regular functions on X (Γ, G)/OutT (G) can be expressed as ratios of polynomials in trace functions. As observed above, the map ϕ : X (Fr, G)/OutT (G) →Ch(Fr, G) is injec- tive generically. However, it is not always injective on X (Fr, G)/OutT (G), see examples in [Vin96]. On the other hand, Proposition 1 shows ϕ is an isomorphism for Γ = Fr and the minimal dimensional algebraic representations of G equal to SL(m, C), Sp(2m, C), or SO(2m + 1, C). It is also an isomorphism for G = SO(2m, C) by [Sik15, Proposition 16]. Question 11. Is

ϕ : X (Fr, G)/OutT (G) →Ch(Fr, G) an isomorphism for any quasi-simple G and any minimal dimensional alge- braic representation of G? The above question is open for groups G other than those mentioned above. In particular, it is open for exceptional groups. (The example of G = SL(2, C) × SO(3, C) of [Vin96] shows the assumption of G being quasi- simple is essential.) Let X sm(Γ, G) and Chsm(Γ, G) denote the smooth loci of X (Γ, G) and Ch(Γ, G), respectively.

Corollary 12. Let Fr be a free group of rank r ≥ 2. If G is a semisimple zd zd subgroup of SL(n, C), then X (Fr, G) is ´etale equivalent to Ch (Fr, G). zd sm Proof. By [FLR16], X (Fr, G) ⊂ X (Fr, G). Since OutT (G) is a fi- zd zd nite group acting freely on X (Fr, G), we conclude that X (Fr, G) → zd zd X (Fr, G)/OutT (G) is ´etale. Since ϕ is a finite, bijective birational map, restricting it to a subspace of the smooth locus gives a local analytic iso- zd zd morphism. Hence X (Fr, G)/OutT (G) → Ch (Fr, G) is also ´etale. Since the composition of ´etale morphisms is ´etale, the result follows. 

We next analyze the groups OutT (G) for quasi-simple G and show that they are often trivial. VARIETIESOFCHARACTERS 9

3. Trace preserving outer automorphisms Out(G) is a subgroup of the symmetries of the Dynkin diagram of the Lie algebra of G (see [FH91, Appendix D]) and, hence, Out(G) is finite for G semisimple. By definition, OutT (G) ⊂ Out(G) and, thus, OutT (G) is trivial whenever Out(G) is trivial.

Example 13. OutT (G) is trivial for every matrix realization of odd orthog- onal groups SO(2m + 1, C), symplectic groups Sp(2m, C), and the complex groups , , , and .

Proof. This follows from the fact that Bn,Cn, G2, F4, E7, and E8 have no symmetries of their Dynkin diagrams, see for example [Sam90].  For m ≥ 3, Out(SL(m, C)) = hσ | σ2i, where σ is the Cartan involution −1 T σ(A) = (A ) . Thus, the group OutT (SL(m, C)) is either trivial or equal to Out(SL(m, C)) for m > 2 depending on whether σ is conjugation by an element of the normalizer of SL(m, C) in the ambient group SL(n, C) . For the canonical matrix realization of SL(m, C) it is easy to see that σ is not trace-preserving for any m> 2. On the other hand, consider the adjoint representation of SL(m, C) which acts irreducibly on

M0 := {M ∈ Mat(m, C) | tr(M) = 0} by conjugation. It is a representation of SL(m, C) into GL(m2-1, C), which we denote by Ad. Example 14. For the standard matrix realization of SL(m, C), the group OutT (SL(m, C)) is trivial. For m ≥ 3,

OutT (Ad(SL(m, C))) =∼ Z/2Z. Proof. Since Out(SL(m, C)) is trivial for m = 2 and tr(σ(A)) 6= tr(A) in general for A ∈ SL(m, C) for m ≥ 3, the first part follows. For the second part, it is enough to prove that there is P ∈ GL(M0) such that σ coincides with conjugation by P via Ad, that is, − P Ad(σ(A))P 1 = Ad(A) for every A ∈ M0. That means − − − P (σ(A)(P 1X)σ(A) 1)= AXA 1 T for every X ∈ M0. Note now that P (M)= M , obviously an invertible linear 2 map, satisfies the above equation. Thus, OutT (Ad(SL(m, C))) = hσ | σ i, as required.  Example 15. For the standard matrix realization of SO(2m, C), the group OutT (SO(2m, C)) is isomorphic to Z/2Z. Proof. Let σ be an automorphism on SO(2m, C) given by conjugation by an orthogonal matrix of determinant −1 as in [Sik15]. Then σ considered as an element of Out(SO(2m, C)) does not depend on the choice of such matrix, [Sik15]. Furthermore, σ2 = Id. It is the only non-trivial outer automorphism 10 S. LAWTON AND ADAM S. SIKORA of SO(2m, C) seen by considering the Dynkin diagram (for m 6= 4). Note that for m = 4 the Dynkin diagram has three symmetry (triality) and so Out(Spin(8, C)) is the on 3 letters, but descending to SO(8, C) reduces the outer automorphisms back down to Z/2Z again. So assuming the defining matrix realization of SO(2m, C), σ preserves the trace 2 and, hence, OutT (SO(2m, C)) = hσ | σ i. 

We note that as shown in [Sam90], for the complex algebraic form of , Out(E6) is generated by an order 2 element as well.

Example 16. There exists a matrix realization of E6 so that OutT (E6) =∼ Z/2Z.

Proof. The minimal dimensional irreducible representation of E6 gives an embedding of E6 into GL(27, C). Then the generator of Out(E6) is, as above, the Cartan involution σ(A) = (A−1)T [Yok09, page 68]. As above, we can then consider the adjoint representation into GL(78, C) which will allow the Cartan involution to be conjugation by an element of the normalizer of E6.  References [AB83] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A, 308(1505):523–615, 1983. [AB94] Norbert A’Campo and Marc Burger. R´eseaux arithm´etiques et commensura- teur d’apr`es G. A. Margulis. Invent. Math., 116(1-3):1–25, 1994. [Ati90] Michael Atiyah. The and physics of knots. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge, 1990. [ATZ95] Helmer Aslaksen, Eng-Chye Tan, and Chen-bo Zhu. Invariant theory of special orthogonal groups. Pacific J. Math., 168(2):207–215, 1995. [BBS13] Kai Behrend, Jim Bryan, and Bal´azs Szendr˝oi. Motivic degree zero Donaldson- Thomas invariants. Invent. Math., 192(1):111–160, 2013. [BFM02] Armand Borel, Robert Friedman, and John W. Morgan. Almost commuting elements in compact Lie groups. Mem. Amer. Math. Soc., 157(747):x+136, 2002. [BH95] G. W. Brumfiel and H. M. Hilden. SL(2) representations of finitely presented groups, volume 187 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1995. [Bul97] Doug Bullock. Rings of SL2(C)-characters and the Kauffman bracket skein module. Comment. Math. Helv., 72(4):521–542, 1997. [BZ98] S. Boyer and X. Zhang. On Culler-Shalen seminorms and Dehn filling. Ann. of Math. (2), 148(3):737–801, 1998. [CCG+94] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen. Plane curves associated to character varieties of 3-manifolds. Invent. Math., 118(1):47–84, 1994. [CG97] Suhyoung Choi and William M. Goldman. The classification of real projective structures on compact surfaces. Bull. Amer. Math. Soc. (N.S.), 34(2):161–171, 1997. [CS83] Marc Culler and Peter B. Shalen. Varieties of group representations and split- tings of 3-manifolds. Ann. of Math. (2), 117(1):109–146, 1983. [Cur01] Cynthia L. Curtis. An intersection theory count of the SL2(C)-representations of the of a 3-manifold. Topology, 40(4):773–787, 2001. VARIETIESOFCHARACTERS 11

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