21 problems

Nonexistence conjecture for the displayed cover. For every finite planar cover

Let $\Sigma$ be a fixed compact surface of genus greater than $1$, let $\mathcal{T}(\Sigma)$ be its Teichmüller space, and let $d_{\mathcal{T}(\Sigma)}$ be the Teichmüller metric.…

Let $\Sigma$ and $\Sigma'$ be closed Riemann surfaces of genus greater than $1$, regarded as Riemannian manifolds with metrics of curvature $-1$. The Ehrenpreis conjecture concerns…

Let $\mathcal{M}$ be a set of Laplace-isospectral manifolds with a uniform upper bound on diameter. Finiteness conjecture. There are only finitely many distinct covering spectra fo…

Let $M$ and $N$ be compact odd-dimensional manifolds, and let $f:M\to N$ be a homotopy equivalence. Let $\widehat M$ and $\widehat N$ be their normal $\Gamma$-covering spaces, and…

Universal covering component conjecture. The connected component of $\tilde A$ in $\Coveringn$ is $\trunc{n+1}A$.

Let $n\in\mathbb{N}$ and $D>0$. A pointed simply-connected proper geodesic space is a pair $(X,p)$, and $\operatorname{Iso}(X)$ denotes its isometry group. For a discrete subgroup…

Let $f:X\rightarrow \mathbb P^1$ be an indecomposable covering of degree $n$, with $X$ of genus $g_X=g$, and let $\operatorname{Mon}(f)$ denote its monodromy group. For each intege…

Liftability conjecture. Every non-split metacyclic action on $S_g$ lifts under a suitably chosen finite regular cyclic cover to a split metacyclic action.

Let $M$ be a finite-volume hyperbolic manifold of dimension at least $4$. Let $f\colon M\to S^1$ be a non-homotopically trivial smooth map such that $\ker(f_*)$ is finitely generat…

Cusp covering conjecture. For each cusp $C$ of $M$, there exists a sublattice $\Lambda_C$ of $\pi_1(C)$ such that, for every choice of a sublattice $\Lambda'_C\subseteq\Lambda_C$ f…

Let $\mathcal{M}$ be a geminal orbit closure, and let the optimal covering map associated with $\mathcal{M}$ be the covering map of minimal degree arising in the classification of…

Let $S$ be a surface and let $S_1$ and $S_2$ be covers of $S$. A closed curve on $S$ has a simple elevation to a cover if one of its elevations is a simple closed curve. Regularity…

Alon and Bilu–Linial conjectures. (1) For every $\epsilon>0$, with probability $1-o(1)$ as $n\to\infty$, every new eigenvalue $\lambda$ of an $n$-cover $X'$ satisfies

Let $t\in\mathbb{N}^*$, let $S^3\subset\mathbb{R}^4$ be the unit $3$-sphere, and let $\nu(5^t)$ denote the integer solutions of … For $x\in\nu(5^t)\cup\nu(5^{t-1})$, write…

Orientation-cover triviality conjecture. This double cover is trivial. The source does not provide a proof or status evidence for this assertion.

Tightness conjecture. For $n\geq 2$,

Let $M$ be a Liouville manifold, let $\widetilde M\to M$ be a cover, and let $W:M\times D^2\to D^2$ be given by $W(m,z)=z^2$. Let $\mathcal F$ denote the relevant Fukaya categories…

Semi-topological inverse Galois conjecture. There exists a Weierstrass polynomial $f$ on $X$ such that the splitting covering $E_f \stackrel{q}{\rightarrow}X$ is equivalent to the…

The Margulis-type cut-off covering spectrum conjecture.

The cut-off covering spectrum conjecture. Theorem 7, namely the corresponding cut-off covering spectrum conclusion established earlier in the paper, holds for such limit spaces $X$…