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

Andrey Egorov

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.

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

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 and . 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 , the maximal dimension where polyhedra of this class exist.

Paper Structure

This paper contains 8 sections, 12 theorems, 20 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds for ideal vertices
  4. Bound for 5-dimensional polyhedra
  5. Bounds for ideal vertices for higher dimensions
  6. Bounds for finite vertices
  7. Bound for 7-dimensional polyhedra
  8. Bounds for finite vertices for higher dimensions

Key Result

Theorem 1.1

Let $P \subset \mathbb{H}^5$ be a right-angled hyperbolic polyhedron of finite volume. Then the number of its ideal vertices $v_\infty(P) \ge 3$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Nikulin--Khovanskii inequality, Nik81Kho86
  • Lemma 2.1: Nonaka, Nonaka
  • Lemma 2.2: Alexandrov, Ale23
  • proof
  • Lemma 2.3: Dufour, Dufour2010
  • Lemma 2.4: Alexandrov, Ale23
  • proof
  • Theorem 3.1
  • ...and 10 more