27 problems

Horizontal-respecting conjecture. Every quasi-isometry

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…

Horizontal-preserving conjecture. Every quasi-isometry $G_M\to G_N$ is horizontal-respecting.

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…

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 $G$ 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 $G\cong \prod_{i=1}^n \mathbf{N}_i \rtimes \mathbb{R}^k$ 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…

Quasi-isometric rigidity conjecture. $S$ and $S'$ are quasi-isometric if and only if they are isomorphic.

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

A purely real Heintze group is a Heintze group $N\rtimes_\alpha\mathbb{R}$ whose defining derivation $\alpha$ has only real eigenvalues. Quasi-isometric rigidity conjecture. Two pu…

Let $X$ and $Y$ be proper, cocompact CAT(0) spaces, and let $f:\partial_*X\to\partial_*Y$ be a homeomorphism, where $\partial_*X$ and $\partial_*Y$ denote their Morse boundaries. A…

Electric-model quasi-isometry conjecture. The map $\varphi$ is a quasi-isometry.

Let $\Gamma$ be a finitely generated group, and let $\rho$ be a linear representation of $\Gamma$ on $\mathbf{R}^n$. Equip $\mathbf{R}^n\rtimes_\rho\Gamma$ and…

A locally compact group is compact-by-Lie if it has a compact normal subgroup such that the quotient is a Lie group. Let $G$ be a compactly generated locally compact group, and let…

Let $G$ be a hyperbolic locally compact group, and let $X$ be a metric space. A group is quasi-isometric to another group or space when their large-scale geometries are equivalent.…

Let $K$ be a global function field, let $S$ be a nonempty set of valuations of $K$, and let $\mathbf{G}$ be a connected, absolutely simple, $K$-isotropic algebraic $K$-group of adj…

Let $K$ be a global function field, let $S$ be a nonempty set of valuations of $K$, and let $\mathbf{G}$ be a connected, absolutely simple, $K$-isotropic algebraic $K$-group of adj…

Let $G$ be a graph that is quasi-isometric to a nonamenable Cayley graph. The quasi-isometric non-Liouville conjecture. The graph $G$ is not Liouville. The source presents this as…