Finite subgroups of the homeomorphism group of a compact topological manifold are almost nilpotent

TL;DR

This paper proves a revised conjecture that finite subgroups of homeomorphism groups of certain topological manifolds contain a nilpotent normal subgroup of bounded index, extending previous results and providing a general strategy for Jordan-type theorems.

Contribution

The paper proves the revised Ghys conjecture for homeomorphism groups of topological manifolds with finitely generated homology, introducing a new approach based on finite group theory.

Findings

Finite subgroups have a nilpotent normal subgroup of bounded index.

Counterexamples to the original conjecture were constructed.

The results extend to non-compact manifolds with finitely generated homology.

Abstract

Around twenty years ago Ghys conjectured that finite subgroups of the diffeomorphism group of a compact smooth manifold M have an abelian normal subgroup of index at most a(M), where a(M) depends only on M. First we construct a family of counterexamples to this conjecture including, for example, the product space $T^2\times S^2$. Following the first appearance of our counterexample on the arXiv Ghys put forward a revised conjecture, which predicts only the existence of a nilpotent normal subgroup of index at most n(M). Our main result is the proof of the revised Ghys conjecture. More generally, we show that the same result holds for homeomorphism groups of not necessarily compact topological manifolds with finitely generated homology groups. Our proofs are based on finite group theoretic results which provide a general strategy for proving similar Jordan-type theorems.