Does Ricci flat with $b_1=0$ imply fundamental group is finite?

Question

Let $(M,g)$ be a Riemannian manifold with Ricci curvature vanishes, that is $\mathrm{Ric}(g)=0$. Now I want to show some topological result about $M$.

Myers' theorem implies if $\mathrm{Ric}(g)>0$, then $\pi_1(M)$ is finite. It's clear that this is false for $\mathrm{Ric}(g)=0$, for example we can consider $n$-torus. But I wonder if we require $b_1(M)=0$, that is first Betti number of $M$ vanishes, does we have $\pi_1(M)$ is finite?

Answer 1

No, the Hantzsche-Wendt manifold is flat, with trivial first Betti number. It is covered by a $3$-torus, so doesn't have finite fundamental group.