13 problems

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…

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

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…

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…

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

Let $G$ be a discrete group with finite asymptotic dimension, and for $g\in G$ let $C_g<G$ denote its centralizer. Centralizer quotient conjecture. The quotient … also has finite a…