On subgroups of finite index in complex hyperbolic lattice triangle groups

TL;DR

This paper investigates finite index subgroups of complex hyperbolic lattice triangle groups, revealing properties like neatness, Betti numbers, and free group homomorphisms, and constructs an infinite tower of neat ball quotients with a single cusp.

Contribution

It provides explicit examples of finite index subgroups with various properties and answers a question about neat ball quotients raised by Stover.

Findings

Some subgroups are neat

Some have positive first Betti number

Existence of homomorphisms onto non-Abelian free groups

Abstract

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.