Several families of incommensurable noncompact hyperbolic Coxeter polytopes

Lizi Guo; Jiming Ma; Yourong Zang; Fangting Zheng

Several families of incommensurable noncompact hyperbolic Coxeter polytopes

Geometric Topology 2026-07-16 v1 Group Theory Number Theory

Authors: Lizi Guo , Jiming Ma , Yourong Zang , Fangting Zheng

Abstract

We classify all 141 finite-volume hyperbolic Coxeter five-dimensional polytopes with eight facets, of which 125 are noncompact. Using maximal-cusp density and a noncompact analog of Bogachev-Douba-Raimbault's argument, we construct infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4, 5, 6, 7, and 9, with the number of commensurability classes growing at least exponentially in volume.

Keywords

hyperbolic Coxeter polytopes noncompact polytopes incommensurable polytopes maximal-cusp density classification

Cite

@article{arxiv.2607.14715,
  title  = {Several families of incommensurable noncompact hyperbolic Coxeter polytopes},
  author = {Lizi Guo and Jiming Ma and Yourong Zang and Fangting Zheng},
  journal= {arXiv preprint arXiv:2607.14715},
  year   = {2026}
}