On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12

TL;DR

This paper establishes new lower bounds for the number of ideal and finite vertices in right-angled hyperbolic polyhedra across dimensions 5 to 12 using geometric and combinatorial methods.

Contribution

It introduces a geometric orthogonal gluing technique and recurrence relations to improve lower bounds in higher dimensions for these polyhedra.

Findings

Lower bound for ideal vertices in 5D: at least 3.

Lower bound for finite vertices in 7D: at least 4.

Bounds are extended up to 12D where such polyhedra exist.

Abstract

We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically $v_\infty(P^5) \ge 3$ and $v_{fin}(P^7) \ge 4$. By combining these initial bounds with double counting arguments and recurrence relations, we obtain improved lower bounds for both types of vertices in all higher dimensions up to $n=12$, the maximal dimension where polyhedra of this class exist.