Can a group G generated by $k$ elements act properly discontinuously freely and cocompactly on $\mathbb{R}^n$, for $n > k$

Question

Equivalently: can the fundamental group of a closed manifold with universal cover homeomorphic to $\mathbb{R}^n$ be generated by $k$ elements, with $k < n$?

As a bonus question: same question without the "free" requirement on the action.

(to be sure: closed = compact + without boundary).

Answer 1

Many closed hyperbolic 3-manifolds have fundamental group generated by 2 elements. Indeed, if you glue two genus 2 handlebodies by a random map of their boundaries the resulting 3-manifold is highly likely to have a hyperbolic structure.

Answer 2

Let $F$ be a free group on 2 generators. Let $N$ be a normal subgroup of index $k-1$ (e.g. the kernel of a homomorphism onto a cyclic group of order $k-1$); then $N$ is free of rank $k$. Then the 2-generated group $F/[N,N]$ is torsion-free (this is an old not-trivial fact) and virtually $\mathbf{Z}^k$ and thus acts isometrically freely properly cocompactly on the Euclidean $k$-space.