4 problems

Virtual infinite Betti number conjecture. A hyperbolic $3$-manifold $M$ has finite covers with arbitrarily large first Betti number; equivalently, for every integer $N>0$, there is…

Waldhausen's conjecture. There is a finite cover $\tilde M\to M$ with $b_1(\tilde M)>0$.

Let $M$ be a compact orientable irreducible atoroidal $3$-manifold with infinite fundamental group. A finite cover of $M$ is a finite-sheeted covering space of $M$, and the first B…

Let $S_g$ be a closed orientable surface and let $\operatorname{Mod}(S_g)$ be its mapping class group. Putman–Wieland conjecture. For $g\geq 3$, no finite-index subgroup of…