Irregular cusps of ball quotients

TL;DR

This paper investigates the boundary structure of ball quotient compactifications, introduces a criterion for general type classification, classifies irregular cusps for certain groups, and explores cusp behavior under embeddings into type IV domains.

Contribution

It introduces the low slope cusp form trick for proving general type and classifies irregular cusps for discriminant kernel subgroups, providing explicit examples.

Findings

Established a criterion for general type of ball quotients.

Classified existence of irregular cusps in specific cases.

Constructed explicit examples of irregular cusps.

Abstract

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.