Immersed turnovers in hyperbolic 3-orbifolds

Abstract

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.

A relative isoperimetric inequality for certain warped product spaces

“…The connection between log convex functions and isoperimetric inequalities appears most recently in explorations of Euclidean space with density (e.g., [8,Corollary 4.11, Theorem 4.13], [2], [3]). The result contained herein is most directly comparable to recent results of Kolesnikov and Zhdanov [6], and generahzes a particular relative isoperimetric inequality for hyperboHc 3-space [7,Section 4].…”

Pants immersed in hyperbolic 3-manifolds

“…This conjecture seems feasible at least when the target manifold is hyperbolic, and the main result in this paper proves this conjecture for immersions of pants. Rafalski [2007] has shown the analogue of this conjecture for turnovers immersed in hyperbolic 3-orbifolds.…”

“…For turnovers immersed in hyperbolic 3-orbifolds, G. Martin [1996] showed that a (2, 3, p)-triangle group in a hyperbolic 3-orbifold is embedded for p ≥ 7. However, there are other triangle groups that are immersed in 3-orbifolds; see [Maclachlan 1996;Rafalski 2007].…”

Small hyperbolic polyhedra

“…Although Theorem 1.3 does not follow from Theorem 1.2, we will provide the proof of the former in the midst of the proof of the latter, as it contains an observation that is necessary for both proofs. The author showed that if a hyperbolic 3-orbifold contains a singular hyperbolic turnover, then that turnover must be contained in a low-volume small 3-suborbifold [11]. In particular, we have the following Moreover, if f is a singular immersion that does not cover an embedded turnover or triangle with mirrored sides, then the component containing f (T ) is unique, and it is a small 3-orbifold.…”

“…Recall from Section 2 that any hyperbolic turnover in a hyperbolic 3-orbifold that does not collapse onto a hyperbolic triangle with mirrored sides may be assumed to be totally geodesic. It also follows from the incompressibility of hyperbolic turnovers in irreducible orbifolds that an immersed turnover must be disjoint from any embedded turnover [11,Lemma 5.3]. Consequently, if T is a hyperbolic turnover, then an immersion f : T → O T lifts to the universal cover H 3 as a collection of geodesic planes with some intersectionstwo or more of these planes will intersect whenever there is a covering transformation (i.e., an element of the fundamental group π 1 (O T ) of O T , which is just the group of isometries of H 3 that yields the quotient O T ) that does not move one plane completely disjoint from some of the others, and this must occur if there is a singular immersion of a turnover in O T -and, additionally, the collection of planes determined by an immersed turnover must be disjoint from the collection of planes determined by any turnover corresponding to a generalized vertex of T .…”