45 problems

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…

Homogeneity Conjecture. The quotient $\Gamma\backslash M$ is homogeneous if and only if every $\gamma\in\Gamma$ is an isometry of constant displacement on $M$.

Let $D6A4$ be a taut graph, let $D6A4^D4$ be the specified family of subgroups, let $M_H$ denote the corresponding covering spaces, and let $D6A4M$ denote the abelian covering of t…

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

Let $X$ and $Y$ be uniform spaces, let $f:X\rightarrow Y$ be a cover, and suppose that $Y$ has uniform dimension at most $n$. Uniform dimension invariance conjecture. The spaces…

Let $X$ be a coverable uniform space with uniform dimension $u\dim X\leq n$, and let $\widetilde{X}$ denote its generalized universal cover. Uniform dimension conjecture. The unifo…

Let $M^n$ be a complete noncompact manifold with nonnegative Ricci curvature. A counterexample to Milnor's conjecture is a manifold whose fundamental group is not finitely generate…

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 $M$ be a connected, orientable open $3$-manifold, let $p:\widetilde{M}\rightarrow M$ be its universal covering map, and let $\gamma$ be a knot in $M$. Suppose that $\pi_1(M)$ i…

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 $\mathcal{M}$ be a class of compact Riemannian manifolds that are Laplace isospectral and have a uniform upper bound on diameter. Finiteness conjecture. There are only finitely…

Let $M$ be a closed, connected, orientable, irreducible $3$-manifold with infinite cyclic fundamental group, and let its universal covering space be denoted by $\widetilde{M}$. Uni…

Let $(M,\nabla_M)$ be a compact and complete affine manifold of dimension $n>2$. A finite cyclic and Galoisian cover is a finite cyclic Galois cover $M'$ of $M$. Finite-cover conje…

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…

Determinant class conjecture. Every normal covering space of a finite simplicial complex is of determinant class.

Let $M$ be an irreducible three-manifold with infinite fundamental group $\pi_1(M)$. A finite covering of $M$ is a covering map $\widetilde M\to M$ of finite degree, and its first…

Let $(X,x_0)=(\mathbb{HA},b)$ with $b\neq 0$, and let … be the map considered in the source. Write $\pi^{sg}(\mathbb{HA},b)$ for the corresponding subgroup of…

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 $d\geq 3$, and let $\mathbf{c}_1,\dots,\mathbf{c}_{d+1}$ be points in general position in $\mathbb{R}^d$. A spray centered in $\mathbf{c}_i$ is a set of the type considered in…

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…

Split-complex covering-map conjecture. A cleaner argument for obtaining the universal covering space may be possible by directly formulating a split-complex covering map analogous…

Let $M$ be a closed hyperbolic $3$-manifold, and let $M_i$ be covering spaces of $M$. Write $\operatorname{spec}(\Delta_N)$ for the spectrum of the Laplacian on a manifold $N$, cou…

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

Aldous–Farb–Sageev conjecture. If $K_1$ and $K_2$ have a common cellular covering, then they have a common finite cellular covering.