Centralizers of reflections in crystallographic groups

Başkan, Tuba Derya; Macbeath, A. M.

doi:10.1017/s0305004100059727

Cited by 10 publications

(31 citation statements)

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“…Consider the centralizer, in the reflection group, of reflection in this face. Baskan and Macbeath [2] proved that the orientation-preserving subgroup of this centralizer is a turnover subgroup of 3 . (A turnover group is the orbifold fundamental group of a turnover.)…”

Section: Then Any Immersion F W T ! Q Of a Hyperbolic Turnover Into Qmentioning

confidence: 98%

Immersed turnovers in hyperbolic 3-orbifolds

Rafalski¹

2010

Groups Geom. Dyn.

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Abstract. We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the "turnover core", whose volume is bounded from above by a function depending only on the area of the given turnover. Furthermore, we show that, for a given type of turnover, there are only finitely many possibilities for the turnover core. As a corollary, if the volume of a complete orientable hyperbolic 3-orbifold is at least 2 and if the fundamental group of the orbifold contains the fundamental group of a hyperbolic turnover (i.e., a triangle group), then the orbifold contains an embedded hyperbolic turnover.Mathematics Subject Classification (2010). 57M50, 57M60.

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Section: Then Any Immersion F W T ! Q Of a Hyperbolic Turnover Into Qmentioning

confidence: 98%

Immersed turnovers in hyperbolic 3-orbifolds

Rafalski¹

2010

Groups Geom. Dyn.

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“…(p) on that hyperbolic plane can be obtained. This applies more generally to any discrete subgroup generated by reflections in the faces of a polyhedron and the structure of the Fuchsian groups so obtained in the cases of the nine tetrahedral groups has been determined (for all this see [1]). …”

Section: Tetrahedral Groupsmentioning

confidence: 99%

“…However, this is the only tetrahedron T { such that a subgroup of the form C£. (ρ) is a triangle group [1].…”

Section: Tetrahedral Groupsmentioning

confidence: 99%

“…Thus if we restrict to orientationpreserving subgroups in both the ambient group and the subgroup, these tetrahedral Kleinian groups will contain Fuchsian subgroups. For only one of the tetrahedra and one of the faces of that tetrahedron, is the Fuchsian subgroup a triangle group [1]. Eight of the tetrahedral groups are arithmetic [20], and in these cases, these tetrahedral groups contain infinitely many commensurablilty classes of Fuchsian subgroups [12].…”

Section: Introductionmentioning

confidence: 99%

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Triangle subgroups of hyperbolic tetrahedral groups

MacLachlan¹

1996

Pacific J. Math.

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It is known that there are nine compact tetrahedra in three-dimensional hyperbolic space for which the group of isometries generated by reflections in the faces is discrete. The subgroup of orientation-perserving isometries in such a group will be called a tetrahedral Kleinian group. Exactly eight such Γ are arithmetic and also a complete list of the finitely many arithmetic Fuchsian triangle groups G is known. In this paper, we determine for which pairs of groups (G, Γ) as above, with one possible exception, one can embed G into Γ. We find that there are many such pairs, contrasting with the single pair (G, Γ) which is known to arise when, instead of arithmeticity, the condition that G be realised as the subgroup of elements of Γ which centralise a reflection in one of the faces of the associated tetrahedron, is imposed.

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“…The subgroup of index 2 consisting of orientation-preserving isometries in these cases is an arithmetic Kleinian group and every such is obtained from a quaternion algebra [4]. These polyhedral groups all contain Fuchsian subgroups [1], and in this paper a result is proved which enables one to identify the quaternion algebra from a quadratic form description of those arithmetic Kleinian groups which contain Fuchsian subgroups.…”

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confidence: 99%