The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary

TL;DR

This paper establishes a polyhedral decomposition for cusped hyperbolic n-manifolds with totally geodesic boundary, extending known decompositions and proving finiteness of such decompositions.

Contribution

It introduces a new polyhedral decomposition for these manifolds, generalizing Epstein-Penner and Kojima decompositions, and proves the finiteness of all such decompositions.

Findings

Existence of a polyhedral decomposition with ideal or partially truncated cells.

Parallel to Epstein-Penner and Kojima decompositions.

Finiteness of the number of such decompositions.

Abstract

Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with totally geodesic boundary. We show that there exists a polyhedral decomposition of $M$ such that each cell is either an ideal polyhedron or a partially truncated polyhedron with exactly one truncated face. This result parallels Epstein-Penner's ideal decomposition \cite{EP} for cusped hyperbolic manifolds and Kojima's truncated polyhedron decomposition \cite{Kojima} for compact hyperbolic manifolds with totally geodesic boundary. We take two different approaches to demonstrate the main result in this paper. We also show that the number of polyhedral decompositions of $M$ is finite.