Non-Discrete Complex Hyperbolic Triangle Groups of Type (m,m,infinity)

TL;DR

This paper proves that certain complex hyperbolic triangle groups of type (m,m,∞) are not discrete when the product of generators is regular elliptic, clarifying conditions for discreteness in complex hyperbolic geometry.

Contribution

It establishes a non-discreteness criterion for (m,m,∞) complex hyperbolic triangle groups based on the nature of the product of generators.

Findings

Groups are not discrete if the product of generators is regular elliptic.

Provides conditions under which (m,m,∞) triangle groups fail to be discrete.

Enhances understanding of discreteness criteria in complex hyperbolic geometry.

Abstract

In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.