48 problems

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

Diestel–Leader conjecture. The graphs $\text{DL}(m,n)$ should provide a negative answer to Woess' question: they are not quasi-isometric to any Cayley graph of a finitely generated…

Brock–Farb's conjecture. The maximal dimension of a quasi-flat in $\mathcal T$ is

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.

Coarse Kuratowski conjecture. Graphs forbidding $K_5$ and $K_{3,3}$ as $K$-fat minors are quasi-isometric to planar graphs.

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…

A purely real Heintze group is a negatively curved homogeneous Lie group of the form $N\rtimes_\alpha\mathbb{R}$, where $N$ is a simply connected nilpotent Lie group and the deriva…

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…

Diestel-Leader non-Cayley conjecture. If $r\ne s$, then $DL(r,s)$ is not quasi-isometric to any Cayley graph.

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 $\mathcal{G}_{(2)}$ be the specified subclass of graphs from the paper, and let $UD_2(\mathsf{\Gamma})$ denote the discrete unordered configuration space of two points on a gra…

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…

Let $J$ be a finite graph and let $K\in\mathbb N$. A graph has no $K$-fat $J$ minor if it contains no geometric $J$ minor whose branch sets and connecting paths are mutually at dis…

Let $K,d\in\mathbb{N}$, and let $H$ be a graph. A graph forbids a $K$-fat $H$ minor if it contains no such fat minor model. Then there exist some $M,A\in\mathbb{N}$ and a graph…

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.

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…

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. By appropriately adjusting the edge lengths of the target graph and modifying the additive distortion constant, the multiplicative distortion facto…

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 $H$ be a finite graph, and let $K$ be a positive parameter. A graph or length space has no $K$-fat $H$ minor if it contains no minor model of $H$ whose branch sets and connecti…

Fat-minor quasi-isometry conjecture. Then $X$ has no $K$-fat $H$ minor for some $K\in\mathbb N$ if and only if $X$ is $f(K)$-quasi-isometric to a graph with no $H$ minor, where…