arXiv subjects

Shawn Rafalski

Publications and source records attributed to Shawn Rafalski.

5 recordsLinked to original sources

arXiv2017

Guts and volume for hyperbolic $3$-orbifolds with underlying space $S^3$

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques involve computing the guts of the orbifold split along the $2$-suborbifold via a careful analysis of its topology. We also characterize the orbifolds of this type that have empty guts.

math.GT↗

arXiv2012

The smallest Haken hyperbolic polyhedra

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.

math.GT↗

arXiv2011

Small hyperbolic polyhedra

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic turnover, we classify the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic tetrahedral orbifolds in the case when all of the vertices of the tetrahedra are non-finite, and we provide a conjectural classification of all the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic polyhedral orbifolds. Finally, we show that any triangle subgroup of a (non-orientable) 3-dimensional hyperbolic reflection group arises from a triangle reflection subgroup.

math.GT↗

arXiv2010

A relative isoperimetric inequality for certain warped product spaces

Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, 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 $k$--dimensional region $F \subset \{b\} \times N$, and the horizontal graph $C \subset \mathbb{R} \times_{f} N$ of an almost everywhere differentiable map over $F$, we prove that the $k$--volume of $C$ is always at least the $k$--volume of the smooth constant height graph over $F$ that traps the same $(1+k)$--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 $k$--volume of $C$ times a constant that depends only on the warping function.

math.GT↗

arXiv2010

Immersed Turnovers In Hyperbolic 3-Orbifolds

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 2πand 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.

math.GT↗