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.

Abstract. We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.

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Given a warped product space M Xf N with logarithmically convex warping function /, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a fc-dimensional region F C {b} x N, and the horizontal graph C cM. Xj N of an almost everywhere differentiable map over F, we prove that the fc-volume of C is always at least the fc-volume of the smooth constant height graph over F that traps the same (1-1-fc)-volume above F as C. We use this to solve a Dido problem for graphs over vertical fibers, and show that, if the warping function is unbounded on the set of horizontal values above a vertical fiber, the volume trapped above that fiber by a graph C is no greater than the fc-volume of C times a constant that depends only on the warping function.

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