# Obstructions for automorphic quasiregular maps and Latt\`es-type uniformly quasiregular maps

## TL;DR

This paper investigates the structure of automorphic quasiregular maps on manifolds, revealing dimensional restrictions based on the properties of their symmetry groups and extending results to Lattès-type maps.

## Contribution

It establishes new dimensional constraints for automorphic quasiregular maps with large translation groups and characterizes Lattès-type maps on various manifolds.

## Key findings

- Dimension of symmetry group is restricted to 0, n-1, or n.
- Strongly automorphic Lattès-type maps have additional restrictions.
- Restrictions are more stringent if the manifold is not a rational cohomology sphere.

## Abstract

Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f \colon \mathbb{R}^n \to M$ 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 Latt\`es-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 Latt\`es case if $M$ is not a rational cohomology sphere.