{"title":"The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary","authors":"Ge Huabin, Jia Longsong, Zhang Faze","doi":"arxiv-2409.08923","DOIUrl":null,"url":null,"abstract":"Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with\ntotally geodesic boundary. We show that there exists a polyhedral decomposition\nof $M$ such that each cell is either an ideal polyhedron or a partially\ntruncated polyhedron with exactly one truncated face. This result parallels\nEpstein-Penner's ideal decomposition \\cite{EP} for cusped hyperbolic manifolds\nand Kojima's truncated polyhedron decomposition \\cite{Kojima} for compact\nhyperbolic manifolds with totally geodesic boundary. We take two different\napproaches to demonstrate the main result in this paper. We also show that the\nnumber of polyhedral decompositions of $M$ is finite.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Geometric Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08923","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.