Finite Heisenberg groups acting on compact manifolds
 
A group G is Jordan if there exists an integer J such that the index of
the maximal abelian subgroup in every finite subgroup of G is at most J,
intuitively, every finite subgroup of G is 'almost' abelian. Ghys
conjectured that the diffeomorphism group of every compact manifold is also
Jordan. This was verified in many cases (e.g. for dimension at most 3, for
non-zero Euler characteristic, for Euclidean spaces and for spheres), but
eventually turned out to be false. The first counterexample is due to
Csikós-Pyber-Szabó (2014) who embedded certain 3-dimensional finite
Heisenberg groups to the diffeomorphism group of S^2xT^2. Based on
this, Mundet i Riera (2014) found many other counterexamples by embedding
higher dimensional finite Heisenberg p-groups satisfying some number
theoretical conditions.

These results raises the following question. Given a family of finite groups
in which the index of the maximal abelian subgroup is not bounded, is there a
compact manifold whose diffeomorphism group contains every member of the
family? A recent result [arxiv.org/abs/1901.07319
<http://arxiv.org/abs/1901.07319>] answers this affirmatively way for: (1)
fixed (but arbitrary) dimensional Heisenberg groups over arbitrary finite
cyclic rings; (2) special p-groups of fixed (but arbitrary) order for every
prime p. The talk focuses on (1).