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Finite groups and complex projective surfaces

Alexander Lubotzky, Matthew Stover

TL;DR

The paper addresses realizing any finite group G as the automorphism group of a closed complex hyperbolic 2-manifold. It develops a construction that starts from cocompact nonarithmetic lattices in PU(2,1) with nontrivial universal homomorphisms and uses congruence-subgroup techniques to produce torsion-free subgroups whose normalizers realize G as a quotient, with Mostow rigidity identifying the automorphism group of the associated manifold with G. A key technical tool is a rigidity result for automorphisms of surface groups, ensuring control over induced automorphisms under the universal-homomorphism framework. The work yields infinitely many isomorphism classes for each G and also translates the results into the setting of irreducible smooth complex projective surfaces of general type, broadening the impact to complex algebraic geometry and the study of Kähler groups.

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 $G$.

Finite groups and complex projective surfaces

TL;DR

The paper addresses realizing any finite group G as the automorphism group of a closed complex hyperbolic 2-manifold. It develops a construction that starts from cocompact nonarithmetic lattices in PU(2,1) with nontrivial universal homomorphisms and uses congruence-subgroup techniques to produce torsion-free subgroups whose normalizers realize G as a quotient, with Mostow rigidity identifying the automorphism group of the associated manifold with G. A key technical tool is a rigidity result for automorphisms of surface groups, ensuring control over induced automorphisms under the universal-homomorphism framework. The work yields infinitely many isomorphism classes for each G and also translates the results into the setting of irreducible smooth complex projective surfaces of general type, broadening the impact to complex algebraic geometry and the study of Kähler groups.

Abstract

In response to a question raised by Belolipetsky and the first author, we prove that for every finite group there are infinitely many isomorphism classes of compact complex hyperbolic -manifolds with automorphism group isomorphic to .
Paper Structure (6 sections, 16 theorems, 44 equations, 2 figures)

This paper contains 6 sections, 16 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices in $\mathop{\mathrm{PU}}\nolimits(2,1)$
  4. Kähler groups
  5. Automorphisms of Fuchsian groups
  6. The main result

Key Result

Theorem 1.1

For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.

Figures (2)

  • Figure 1: A compact elliptic fibration $E(n)$
  • Figure 2: The relationships between various groups

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 23 more