5 problems

Putman–Wieland conjecture. For every $g\ge 2$, every $p\ge 0$, and every finite-index subgroup $K<\pi_1(\Sigma^p_g,\star)$, the subgroup $K$ has the Putman–Wieland property.

Uniform spectral gap conjecture. For any $\varepsilon>0$, with probability tending to $1$ as $n\to +\infty$, the cover $M_n\to M$ has no new resonance in the half-space

Let $X$ be a closed negatively curved surface with universal cover $\widetilde{X}$, and let $\lambda_0=\lambda_0(\widetilde{X})$ be the bottom of the $L^2$-spectrum of the Laplacia…

Let $X$ be a closed hyperbolic surface, and let $X_\phi$ be a generic covering surface of $X$ whose covering degree tends to infinity. For every $\epsilon>0$, consider the spectrum…

Random-cover spectral-gap conjecture. For every $\varepsilon>0$, with probability tending to $1$ as $n\to\infty$, there are no new eigenvalues of $X_n$ below…