Classification of complex hyperbolic triangle groups by types

TL;DR

This paper provides a comprehensive classification of complex hyperbolic triangle groups based on the ellipticity of specific words, advancing the understanding of their structure and confirming aspects of the Schwartz conjecture.

Contribution

It offers a complete classification of complex hyperbolic triangle groups by types, refining previous results and building on the proof of the Schwartz conjecture.

Findings

Classification of groups by ellipticity of key words

Improvement over previous partial classifications

Confirmation of the Schwartz conjecture aspects

Abstract

We give a complete classification of complex hyperbolic $(n_1, n_2, n_3)$-triangle groups by types defined according to the ellipticity of two particular words of short length. This improves the Schwartz conjecture proved by Grossi.