Volumes of Picard modular surfaces

TL;DR

This paper identifies the smallest volume arithmetic complex hyperbolic 2-orbifolds and classifies minimal volume manifolds covering them, advancing understanding of hyperbolic geometry and orbifold classification.

Contribution

It proves the minimal volume cusped complex hyperbolic 2-orbifolds are the two smallest arithmetic examples and classifies all minimal volume manifolds covering these orbifolds.

Findings

The two smallest arithmetic complex hyperbolic 2-orbifolds are the minimal volume cusped examples.

Every minimal volume arithmetic cusped complex hyperbolic 2-manifold covers one of these orbifolds.

All minimal volume manifolds covering both orbifolds are explicitly described.

Abstract

We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that simultaneously cover both minimal orbifolds.