Immersed Turnovers In Hyperbolic 3-Orbifolds

TL;DR

This paper investigates the structure of immersions of hyperbolic turnovers in 3-orbifolds, establishing bounds on associated suborbifolds and finiteness results, with implications for the presence of embedded turnovers in certain orbifolds.

Contribution

It introduces the concept of the 'turnover core' and proves volume bounds and finiteness results for hyperbolic 3-orbifolds containing turnovers, advancing understanding of their geometric structure.

Findings

Immersed turnovers are contained in a bounded-volume hyperbolic suborbifold.

Finitely many turnover cores exist for each turnover type.

Orbifolds with volume ≥ 2π containing a turnover group have embedded turnovers.

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\pi 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.