Groups, Geometry, and Dynamics

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Volume 4, Issue 2, 2010, pp. 333–376
DOI: 10.4171/GGD/86

Published online: 2010-02-21

Immersed turnovers in hyperbolic 3-orbifolds

Shawn Rafalski[1]

(1) Fairfield University, USA

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 2π 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.

Keywords: Hyperbolic 3-orbifold, triangle group, hyperbolic turnover, hyperbolic volume, immersed suborbifold

Rafalski Shawn: Immersed turnovers in hyperbolic 3-orbifolds. Groups Geom. Dyn. 4 (2010), 333-376. doi: 10.4171/GGD/86