Discrete embeddings of hyperbolic groups with Pontryagin-surface boundaries

Abstract

Let $X_p$ be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2\pi /p$. For every $2\le p\le 8$, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to $X_p$ that admits a discrete, faithful, convex cocompact reflection representation into Isom$({\mathbf{H}}^5)$, whose limit set is homeomorphic to the index-$p$ Pontryagin surface ${\Pi }_p$. Dimension five is optimal, since ${\Pi }_p$ does not embed in ${\mathbb{S}}^3$. For $p=2,3$, the constructions are analytic and yield cyclically symmetric infinite families.

40 pages