On the character variety of group representations in SL(2, ℂ) and PSL(2, ℂ)

González-Acuña, F.; Montesinos-Amilibia, José María

doi:10.1007/bf02572429

Cited by 75 publications

(106 citation statements)

References 8 publications

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“…Section 4 provides the finishing touch. We rely on a parameterization of the character set given in [8]. Most of the defining polynomials turn out to be specialized Procesi identities, while the remaining few succumb to other tools from Section 3.…”

Section: Theorem If M Contains No Non-boundary Parallel Incompressimentioning

confidence: 99%

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Bullock¹

1997

Comment. Math. Helv.

128

157

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Abstract. Let M be a compact orientable 3-manifold. The set of characters of SL 2 (C )-representations of π 1 (M ) forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional.Mathematics Subject Classification (1991). 57M99.

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Section: Theorem If M Contains No Non-boundary Parallel Incompressimentioning

confidence: 99%

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Bullock¹

1997

Comment. Math. Helv.

128

157

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“…In 1983, Culler and Shalen defined the character variety and showed that it is in fact an algebraic set [6]; the set is the image under a "trace" map. González-Acuña and Montesinos-Amilibia showed in 1993 that the relations of Magnus in fact determine the algebraic set that Culler and Shalen had defined [19]. In 2001, Sikora, using results of Procesi [26], showed that the character variety of SL(n, C) can be realized as spaces of graphs subject to topologically motivated relations [30].…”

Section: History Of Sl(2 C) Invariant Theorymentioning

confidence: 99%

Handbook of Teichmüller Theory, Volume II

Lawton¹,

Peterson²

2009

IRMA Lectures in Mathematics and Theoretical Physics

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“…(see González-Acuña and Montesinos-Amilibia's paper [12]). Now, from the presentation (6) above it is clear that each representation ρ : π 1 (M, * ) → PSL 2 C lifts to exactly two representation ±ρ :…”

Section: Examplesmentioning

confidence: 99%

“…(These points correspond to the trivial representation and to the two orientation-preserving conjugacy classes of discrete and faithful representations.) Since the line V = 2 corresponds to reducible representations (see [12]) Orth K takes the whole line to the point (1, 1). On the curve (9), Proposition 12 implies that…”

Section: Examplesmentioning

confidence: 99%

A new invariant on hyperbolic Dehn surgery space

Dowty

2002

Algebr. Geom. Topol.

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In this paper we define a new invariant of the incomplete hyperbolic structures on a 1-cusped finite volume hyperbolic 3-manifold M , called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of M which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in M . Conjecturally this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery space of M .

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On the character variety of group representations in SL(2, ℂ) and PSL(2, ℂ)

Cited by 75 publications

References 8 publications

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Handbook of Teichmüller Theory, Volume II

A new invariant on hyperbolic Dehn surgery space

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