Effective bounds for Vinberg's algorithm for arithmetic hyperbolic lattices

TL;DR

This paper enhances Vinberg's semi-algorithm for hyperbolic reflection groups by providing an effective termination condition, enabling it to find maximal reflection sublattices in arithmetic hyperbolic lattices.

Contribution

It introduces an effective termination criterion for Vinberg's semi-algorithm, making it a practical algorithm for identifying maximal reflection sublattices.

Findings

Established an upper bound for faces of hyperbolic Coxeter polyhedra based on volume.

Converted Vinberg's semi-algorithm into a complete algorithm with termination guarantees.

Demonstrated the effectiveness of the new bounds in hyperbolic lattice computations.

Abstract

A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.