Obstructions for automorphic quasiregular maps and Lattès-type uniformly quasiregular maps

Abstract

Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f:{\mathbb{R}}^n\to M$ is a quasiregular map automorphic under a discrete group $\Gamma$ of Euclidean isometries, and $f$ has finite multiplicity in a fundamental cell of $\Gamma$. We show that if $\Gamma$ has a sufficiently large translation subgroup ${\Gamma }_T$, then $\dim \Gamma \in \{0,n-1,n\}$. If $f$ is strongly automorphic and induces a non-injective Lattès-type uniformly quasiregular map, then the same holds without the assumption on the size of ${\Gamma }_T$. Moreover, an even stronger restriction holds in the Lattès case if $M$ is not a rational cohomology sphere.

Version 2 includes a short yet significant improvement to the original results, pointed out to the author by Sylvester Eriksson-Bique

References in corpus (1)

• Sharp cohomological bound for uniformly quasiregularly elliptic manifolds