Large and iterated finite group actions on manifolds admitting non-zero degree maps to nilmanifolds

TL;DR

This paper investigates finite group actions on manifolds that map to nilmanifolds with non-zero degree, establishing bounds on symmetries, studying stabilizers, and exploring cohomological rigidity for certain nilmanifold classes.

Contribution

It introduces the concepts of free iterated group actions and iterated discrete degree of symmetry, providing new insights into symmetry bounds and rigidity for these manifolds.

Findings

Homeo(M) is Jordan for such manifolds

Bounded the discrete degree of symmetry of M

Established cohomological rigidity results for 2-step nilmanifolds

Abstract

Let M be a closed connected oriented manifold admitting a non-zero degree map to a nilmanifold. In the first part of the paper we study effective finite group actions on M. In particular, we prove that Homeo(M) is Jordan, we bound the discrete degree of symmetry of M and we study the number and the size of stabilizers of an effective action of a finite group G on M. We also study the toral rank conjecture and Carlsson's conjecture for large primes for this class of manifolds. In the second part of the paper we introduce the concepts of free iterated action of groups and the iterated discrete degree of symmetry which we use to obtain cohomological rigidity results for manifolds admitting a non-zero degree map to a 2-step nilmanifold.