# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.07319