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

Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to M$ is a quasiregular map automorphic under a discrete group $\mathrm{\Gamma }$ of Euclidean isometries, and $f$ has finite multiplicity in a fundamental cell of $\mathrm{\Gamma }$. We show that if $\mathrm{\Gamma }$ has a sufficiently large translation subgroup ${\mathrm{\Gamma }}_T$, then $\dim \Gamma \in {0, n-1, n}$\mathrm{.}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 ${\mathrm{\Gamma }}_T$. Moreover, an even stronger restriction holds in the Latt`es case if $M$ is not a rational cohomology sphere.