Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups

A generalised triangle group has a presentation of the form where R is a cyclically reduced word involving both x and y. When R=xy, these classical triangle groups have representations as discrete groups of isometries of S 2 , R 2 , H 2 depending on In this paper, for other words R, faithful discrete representations of these groups in Isom + H 3 = PSL(2,C) are considered with particular emphasis on the case /? = [x, y] and also on the relationship between the Euler characteristic x and finite covolume representations.1991 Mathematics subject classification: 20H15.

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“…[23,18] In the arithmetic cases, kF <2) and AF {2) coincide with the defining field and quaternion algebra.…”

Section: Arithmeticitymentioning

confidence: 99%

On discrete generalised triangle groups

Hagelberg

MacLachlan

Rosenberger

1995

Proceedings of the Edinburgh Mathematical Society

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“…In particular, F 4 contains no closed, embedded, totally geodesic surface. Even better, it is known [9], [11] that Γ m is arithmetic for m = 4, 5, 6, 8, and 12 and the results of [19] applied to the case m = 4 show that F 4 contains no non-elementary Fuchsian subgroups at all (the invariant trace field and quaternion algebra for Γ 4 are computed in [27]). Our result should also be contrasted with the fact that the complex structure on Γ\SL(2, C) is rigid if and only if the first Betti number of Γ is zero [6], [26].…”

Section: Discussionmentioning

confidence: 99%

Infinitesimal deformations of some SO(3,1) lattices

Scannell¹

2000

Pacific J. Math.

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Let Γ be a torsion-free lattice in SO 0 (3, 1), and let M = Γ\H 3 be the corresponding hyperbolic 3-manifold. It is wellknown that in the presence of a closed, embedded, totallygeodesic surface in M , the canonical flat conformal structure on M can be deformed via the bending construction. Equivalently, the lattice Γ admits non-trivial deformations into SO 0 (4, 1). We present a new construction of infinitesimal deformations for the hyperbolic Fibonacci manifolds, the smallest of which is non-Haken and contains no immersed totally geodesic surface.

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“…Theorems 4 and 5 of [21]). We now establish the following characterization of arithmetic Jørgensen groups of elliptic type, which completes the proof of Theorem 1.5 as a corollary.…”

Section: Theorem 52 a Finite-covolume Kleinian Group γ Is Arithmetimentioning

confidence: 99%

Jørgensen number and arithmeticity

Callahan

2009

Conform. Geom. Dyn.

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Abstract. The Jørgensen number of a rank-two non-elementary Kleinian group Γ isJørgensen's Inequality guarantees J(Γ) ≥ 1, and Γ is a Jørgensen group if J(Γ) = 1. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper concludes with computations of J(Γ) for several noncocompact Kleinian groups including some two-bridge knot and link groups.

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Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups

Cited by 64 publications

References 7 publications

On discrete generalised triangle groups

On discrete generalised triangle groups

Infinitesimal deformations of some SO(3,1) lattices

Jørgensen number and arithmeticity

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