From Geometry to Arithmetic of Compact Hyperbolic Coxeter Polytopes

Abstract

In 1974, Kaplinskaja classified all simplicial straight hyperbolic Coxeter prisms. In this paper, we determine precisely which of these prisms are properly quasi-arithmetic or arithmetic. We also present some observations regarding commensurability classes and systoles of the associated orbifolds. Finally, we show that there is a cocompact properly quasi-arithmetic reflection group in H 3 preserving an isotropic quadratic form. This phenomenon was previously known due to Vinberg only in dimension 2.

Effective bounds for Vinberg’s algorithm for arithmetic hyperbolic lattices

“…We refer to [3] for the references regarding implementations and applications of the algorithm. More recent references are [6][7][8] (a theoretical description of the algorithm, its Python implementation, and some applications), [16, §6.2.1],…”