4 problems

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 $S_1$ and $S_2$ be simply connected solvable Lie groups of real type, meaning that all eigenvalues of $\operatorname{ad}(X)$ are real for every $X$ in their Lie algebras. Cornu…

Let $S$ be a SOL-like group. Suppose $\operatorname{Isom}(S,g)$ does not admit a uniform lattice for any left invariant Riemannian metric $g$ on $S$. The non-lattice conjecture. Th…

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