40 problems

Fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}\to\mathbb{N}^2$ such that, for every graph $G$ and $K\in\mathbb{N}$, if $G$ does not contain $X$ as…

Let $\mathbf{B}_2(\mathbb{Z}[\sqrt{2}])$ be the indicated polycyclic group, which is a discrete cocompact subgroup of the three-dimensional solvable Lie group $\operatorname{Sol}$.…

Margulis' conjecture. There is an isometry $\tilde f:X\longrightarrow Y$ at bounded distance from $f$.

Let $M,N\in\operatorname{SL}(n,\mathbf{Z})$ lie on $1$-parameter subgroups of $\operatorname{GL}(n,\mathbf{R})$, and suppose that neither $M$ nor $N$ has an eigenvalue on the unit…

Let $G$ be a solvable Lie group, and let $\Gamma$ be a lattice in $G$. A group is virtually a lattice if it is weakly commensurable to a lattice, where weak commensurability is gen…

Let $G$ and $H$ be finite groups, and let $G\wr\mathbb{Z}$ and $H\wr\mathbb{Z}$ be their lamplighter groups. Conjecture I. Then $G\wr\mathbb{Z}$ is quasi-isometric to…

Let $G$ be a nontrivial finite group, let $G\wr\mathbb{Z}=(\bigoplus_{\mathbb{Z}}G)\rtimes\mathbb{Z}$ be its lamplighter group, and let $G((t))$ denote the associated boundary spac…

Let $\mathbb{F}_q((t))$ be the Laurent-series field over the finite field $\mathbb{F}_q$, and let $\operatorname{Bilip}(\mathbb{F}_q((t)))$ denote its bilipschitz self-homeomorphis…

Let $\mathbb{F}_q$ be a finite field, let $\mathbb{F}_q((t))$ be its Laurent-series field, and let $\mathbb{F}_q[t,t^{-1}]$ be the Laurent-polynomial ring. For $n\geq 3$, let…

Classification conjecture. The groups $\Gamma_M$ and $\Gamma_N$ are quasi-isometric if and only if there exist nonzero $a,b\in\mathbb R$ such that $M^a$ and $N^b$ have the same abs…

Let $M\in\operatorname{SL}(n,\mathbf{Z})$ have no eigenvalues on the unit circle, and let $G$ be a finitely generated group quasi-isometric to the associated group $\Gamma_M$. The…

Let $G$ be a geometrically rigid hyperbolic group. An abstract commensurator of $G$ is an isomorphism $\phi:H_1\to H_2$ between finite-index subgroups $H_1,H_2\leq G$. A map…

Let $G$ and $H$ be simply connected nilpotent Lie groups. Losert–Malcev conjecture. The groups $G$ and $H$ are quasi-isometric if and only if they are isomorphic. This would mean t…

Let $\mathbb G=\mathbb F\rtimes_\phi\mathbb Z$ and $\mathbb G'=\mathbb F'\rtimes_\psi\mathbb Z$ be free-by-cyclic groups. Let $\mathfrak d(\phi)$ denote the length, or size, of a l…

Let $\mathbb G=\mathbb F\rtimes_\phi\mathbb Z$ and $\mathbb G'=\mathbb F'\rtimes_\psi\mathbb Z$ be free-by-cyclic groups, where an outer automorphism is fully irreducible if it has…

Let $\mathbb G$ and $\mathbb G'$ be quasi-isometric free-by-cyclic groups. A slender splitting is a splitting of a group over a slender subgroup. Slender-splitting conjecture. A sl…

Let $G\cong \prod_{i=1}^n \mathbf{N}_i \rtimes \mathbb{R}^k$ be nondegenerate with $n\ge 2$. Let $q:G\to G$ be a quasi-isometry for some, and hence any, left-invariant Riemannian m…

Let $G$ and $H$ be finitely generated groups, and write $\operatorname{pdim}(G)$, $\nu\dim(G)$, and ${}^{e}\operatorname{pdim}(G)$ for the three percolation-dimension notions intro…

Let $M= obreakbackslash mathbb{H}^n$ be a closed hyperbolic $n$-manifold with fundamental group $Gamma$, where either $nleq 3$ or $M$ is arithmetic of simplest type. A **$lambda$-q…

A graph $\Gamma$ is minor-excluded if some finite graph is not a minor of $\Gamma$. Two graphs are quasi-isometric when their large-scale metric spaces are equivalent up to multipl…

A graph $\Gamma$ is minor-excluded if some finite graph is not a minor of $\Gamma$. A graph is accessible if it admits a finite decomposition over finite separators with pieces hav…

Cornulier's conjecture. There exist negatively curved homogeneous spaces other than symmetric spaces that are quasi-isometric to finitely generated groups.

Generalized collapsing-map conjecture. Then the generalized form of Theorem 2 and Corollary 2.1 holds.

Factorization conjecture. Any quasi-isometry can be expressed as the composition of a collapsing map and Lipschitz map.

Scaling-group conjecture. The scaling group of $G$ is a subgroup, possibly trivial, of