Rigidity of extremal quasiregularly elliptic manifolds

TL;DR

This paper characterizes closed manifolds admitting quasiregular maps from Euclidean space, showing their fundamental groups are virtually free abelian of rank n if and only if the manifold is aspherical and has growth order n.

Contribution

It establishes an equivalence between growth order, asphericity, and the structure of the fundamental group for quasiregularly elliptic manifolds.

Findings

Fundamental group growth order equals manifold dimension.

Manifolds are aspherical if they admit such quasiregular maps.

Fundamental groups are virtually Z^n and torsion free.

Abstract

We show that for a closed $n$-manifold $N$ admitting a quasiregular mapping from the Euclidean $n$-space the following are equivalent: (1) order of growth of $\pi_1(N)$ is $n$, (2) $N$ is aspherical, and (3) $\pi_1(N)$ is virtually $\mathbb{Z}^n$ and torsion free.