26 problems

Let $(M,g)$ be a complete, noncompact manifold with finitely many ends, each end outside a compact set diffeomorphic to a spherical shell. Paraboloid-end conjecture. Each of the sp…

Coarse separability conjecture. The graph $G$ has a disconnecting clique if and only if the graph product $\Gamma(G)$ is coarsely separable by a family of subexponential growth.

A length space is a metric space in which the distance between two points is the infimum of the lengths of paths joining them. A length space is quasi-isometric to a graph when the…

The $k$-planar conjecture. The graph $\Gamma$ is almost planar if and only if it is quasi-isometric to some planar graph.

Let $G$ and $H$ be quasi-isometric groups of type $\FP_n(R)$, where $R$ is an arbitrary coefficient ring. Write $\operatorname{FV}_{R,G}^{n}$ and $\operatorname{FV}_{R,H}^{n}$ for…

Ultra-fat half-grid conjecture. Every connected, quasi-transitive, locally finite graph with a thick end contains an ultra-fat model of the half-grid.

Quasi-isometry classification conjecture. The groups $W_\Gamma$ and $W_{\Gamma'}$ are quasi-isometric if and only if they are isomorphic.

Let $d\geq 2$ be an integer, and consider the class of graphs with sphere dimension at most $d$. Quasi-isometry conjecture. This class is quasi-isometric to the class of intersecti…

Edge-contraction and subdivision conjecture. For all $L,C\in\mathbb{N}$, there exists $C'\in\mathbb{N}$ such that if a graph $G$ is $(L,C)$-quasi-isometric to a graph in…

Nguyen–Scott–Seymour conjecture. For all $L,C\in\mathbb{N}$, there exists $C'\in\mathbb{N}$ such that if $\varphi$ is an $(L,C)$-quasi-isometry from a graph $G$ to a graph $H$, the…

Nguyen–Scott–Seymour conjecture. There is a constant $k$ such that if a graph $G$ admits a quasi-isometry to a graph of tree-width at most two, then $G$ admits a quasi-isometry wit…

Kuratowski-type coarse planarisation conjecture. There exists a function $f$ such that every graph with no $K$-fat $K_5$ and $K_{3,3}$ minor is $f(K)$-quasi-isometric to a planar g…

Quasi $n$-bottlenecking conjecture. If $G$ is coarsely $n$-bottlenecked, then it is quasi-isometric to an $n$-edge bottlenecked graph.

Quasi-isometry invariance conjecture. Then

Let $S_1$ and $S_2$ be simply connected solvable Lie groups of real type, meaning that all eigenvalues of $\operatorname{ad}(X)$ are real for every $X$ in their Lie algebras. Cornu…

Let $S$ be a SOL-like group. Suppose $\operatorname{Isom}(S,g)$ does not admit a uniform lattice for any left invariant Riemannian metric $g$ on $S$. The non-lattice conjecture. Th…

Fix, for every integer $n\geq 1$, the fundamental group $G_n$ of an $(n+2)$-dimensional cusped hyperbolic manifold. Let $\bar n$ be a finite collection of integers at least $1$, an…

Let $G$ be a finitely generated nilpotent group, and for each $i$ let $r_i$ be the torsion-free rank of the quotient $G^{(i+1)}/G^{(i)}$. Nilpotent conjugacy-growth rank conjecture…

Let $G$ and $H$ be simply connected nilpotent Lie groups. Quasi-isometric rigidity conjecture. The groups $G$ and $H$ are quasi-isometric if and only if they are isomorphic. This w…

Quasi-isometric invariance of irreducibility and atoroidality. If $F\rtimes_\phi\mathbb Z$ and $F'\rtimes_\psi\mathbb Z$ are quasi-isometric, then $\phi$ is irreducible and atoroid…

Let $A=A_S$ be an irreducible Artin–Tits group of spherical type with rank at least $3$. Let $X_{abs}^A$ be the absorbable-element generating set, $X_{NP}^A$ the union of normalize…

Let $N$ and $M$ be two closed graph manifolds. Let $S \looparrowright N$ and $S' \looparrowright M$ be two separable, horizontal surfaces. Quasi-isometry conjecture. There is a qua…

Quasi-isometry–commability conjecture. The group $H$ is commable to $G$.

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. The cell-wise first-wave critical expone…

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. A critical exponent is the power-law exp…