22 problems

Let $G$ be a hyperbolic group, let $\partial_{\infty}G$ denote its boundary at infinity, let $\operatorname{asdim} G$ denote its asymptotic dimension, and let…

Let $G$ be a group that is weakly hyperbolic relative to a finite collection of subgroups $\{H_\lambda\}_{\lambda\in\Lambda}$. Assume that each subgroup $H_\lambda$ has finite asym…

Let $n\geq 2$ be a positive integer, and let $\mathcal{F}$ be the class of intersection graphs of families of spheres in $\mathbb{R}^n$. The asymptotic dimension of a graph class i…

For a graph $F$, a graph class excludes $F$ as an induced minor when none of its graphs contains $F$ as an induced minor. A graph class has finite asymptotic dimension when its asy…

Let $n$ be a positive integer, and let $A$ be a subspace of $\mathbb{R}^n$. The asymptotic dimension of $A$ is denoted by $\operatorname{asdim}(A)$, and $A$ separates…

Asymptotic-dimension conjecture.

Let $d\geq 1$ be an integer, and consider the class of graphs whose sphere dimension is at most $d$. Asymptotic-dimension conjecture. This class has asymptotic dimension at most…

An $n$-asymptotic Cantor manifold is a metric space of asymptotic dimension $n$ that cannot be coarsely separated by a subset of asymptotic dimension at most $n-2$. For each intege…

Let $F$ be a nonabelian free group, and let $\operatorname{pdim}(F)$ denote its periodic asymptotic dimension. Free-group periodic-dimension conjecture. Nonabelian free groups have…

Rank-one symmetric-space conjecture. For every such $X$ and every $d$,

Subpolynomial asymptotic-dimension conjecture.

Let $X$ be a symmetric space with trivial compact factor, and let $\ell\in\mathbb{N}$. Here $T_3$ denotes the regular tree of valence $3$, and $\operatorname{corank}(X)$ is the cor…

Tselekidis's conjecture. The asymptotic dimension of the Coxeter group satisfies

Tselekidis's conjecture. The inequality

Let $X$ be a fuzzy metric space, and let $n$ be a nonnegative integer. Conditions,, and in Theorem are the three conditions stated there for $X$ and $n$. Alternative-characterizati…

Let $\mathcal D_\infty=\varinjlim \mathcal D_n$, where each $\mathcal D_n$ is endowed with a Wasserstein metric; one may also consider the Wasserstein $p,q$ metric. Wasserstein asy…

Singleton-corona conjecture. If the asymptotic dimension of $LS(\omega)$ does not equal $0$, then there is an unbounded subset $C$ of $X$ whose corona consists of exactly one point…

Asymptotic-dimension conjecture. The hyperballean of $G$ has infinite asymptotic dimension:

Finite asymptotic dimension conjecture. Every geodesic coarse median space with finite rank has finite asymptotic dimension.

The remetrization of a metric space $(X,\rho)$ given by $\rho'(x,y)=\ln(1+\rho(x,y))$ relates asymptotic power dimension to asymptotic Assouad–Nagata dimension. Remetrization conje…

Let $S$ be a surface, and let $\operatorname{Mod}(S)$ denote its mapping class group. Write $\operatorname{asdim}\operatorname{Mod}(S)$ for its asymptotic dimension,…

Let $X$ be a metric space, and let $\operatorname{trasdim}X$ denote its transfinite asymptotic dimension. Omega conjecture. If … then $\operatorname{trasdim}X=\omega$. The conjectu…