Diffeomorphism Groups of Compact 4-manifolds are not always Jordan

TL;DR

The paper provides a counterexample to Ghys's conjecture by showing certain 4-manifolds have diffeomorphism groups lacking the Jordan property, highlighting complex symmetry structures in these manifolds.

Contribution

It demonstrates that the diffeomorphism groups of specific compact 4-manifolds do not satisfy the Jordan property, countering a longstanding conjecture.

Findings

Diffeomorphism groups of certain 4-manifolds lack the Jordan property.

Constructs explicit finite subgroups with unbounded abelian subgroup indices.

Provides a counterexample to Ghys's conjecture.

Abstract

We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.