An Alternative to Vinberg's Algorithm

TL;DR

This paper analyzes the slowness of Vinberg's algorithm for hyperbolic reflection groups, proves its limitations, and introduces a faster alternative based on quadratic form vector searches.

Contribution

It provides a theoretical explanation for Vinberg's algorithm's slowness and proposes a novel, more efficient algorithm leveraging quadratic form vector computations.

Findings

Vinberg's algorithm can be inherently slow in certain cases.

The new algorithm significantly improves computational speed.

The approach involves finding small positive norm vectors in quadratic forms.

Abstract

Vinberg's algorithm is the main method for finding fundamental domains for reflection groups acting on hyperbolic space. Experience shows that it can be slow. We explain why this should be expected, and prove this slowness in some cases. And we provide an alternative algorithm that should be much faster. It depends on an algorithm for finding vectors with small positive norm in indefinite binary quadratic forms, of independent interest.