Special $p$-groups acting on compact manifolds

TL;DR

This paper constructs compact manifolds whose diffeomorphism groups contain all special p-groups of a given order, providing counterexamples to the Jordan property conjecture for diffeomorphism groups.

Contribution

It generalizes previous work by constructing manifolds with diffeomorphism groups containing all special p-groups of a fixed order, regardless of exponent.

Findings

Constructed manifolds with rich diffeomorphism groups containing all special p-groups of order p^r.

Provided explicit counterexamples to Ghys's conjecture on the Jordan property of diffeomorphism groups.

Extended the class of groups known to act on compact manifolds beyond previous results.

Abstract

Riera proved at arXiv:1412.6964 that the diffeomorphism group of particular compact manifolds are not Jordan by exhibiting subgroups isomorphic to extra-special $p$-groups of exponent $p$ for primes $p$ satisfying some conditions. Generalising the methods of that paper, we construct a compact connected smooth real manifold for every natural number $r$ whose diffeomorphism group contains not only every extra-special $p$-group, but also every special $p$-group of order $p^r$ independently of its exponent for every prime $p$. We obtain a similar statement about finite Heisenberg groups as well as we display a very explicit counterexample to the conjecture of Ghys about Jordan property of diffeomorphism group of compact manifolds.