Finite groups and complex projective surfaces

TL;DR

This paper proves that for any finite group, there exist infinitely many compact complex hyperbolic 2-manifolds whose automorphism groups are isomorphic to that group, addressing a question in the field.

Contribution

It establishes the existence of infinitely many such manifolds for every finite group, expanding understanding of symmetries in complex hyperbolic geometry.

Findings

For every finite group, infinitely many isomorphism classes of manifolds exist.

Automorphism groups of these manifolds can be precisely controlled.

Addresses a previously open question in complex hyperbolic geometry.

Abstract

In response to a question raised by Belolipetsky and the first author, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.