Line arrangements in $\mathbb {H}^3$

Abstract:

If $M=\mathbb {H}^3/G$ is a hyperbolic manifold and $\gamma \subset M$ is a simple closed geodesic, then $\gamma$ lifts to a collection of lines in $\mathbb {H}^3$ acted upon by $G$. In this paper we show that such a collection of lines cannot contain a particular type of subset (called a bad triple) unless $G$ has orientation-reversing elements. This fact allows us to extend certain lower bounds on hyperbolic volume to the non-orientable case.