Geometric rigidity of quasi-isometries in horospherical products
Tom Ferragut
TL;DR
This work proves geometric rigidity for quasi-isometries between horospherical products $X\bowtie Y$ of Gromov hyperbolic, Busemann spaces under the condition $m>n$ (and similarly for the primed pair). By developing a robust toolkit—ε-monotone quasigeodesics, coarse differentiation, height-respecting tetrahedra, and precise box tilings with horospherical measures—the authors show that any quasi-isometry $\Phi:X\bowtie Y\to X'\bowtie Y'$ is uniformly close to a product map $\left(\Phi^X,\Phi^Y\right)$ (up to permutation of factors). They introduce horo-admissible measures $(X,a,\mu^X)$ and $(Y,b,\mu^Y)$ and a derived measure $\mu$ on $X\bowtie Y$, along with a height-respecting projection framework, to derive quasi-isometry invariants such as $\frac{m}{n}$ and, in the solvable Lie group setting, a complete quasi-isometry classification for Carnot–Sol type groups. The results extend Eskin–Fisher–Whyte’s rigidity to a broad class of horospherical products, enabling new quasi-isometric classifications of solvable Lie groups and a detailed picture of the quasi-isometry group in these geometries.
Abstract
We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in [7]. Our work covers the case of solvable Lie groups of the form R ___ (N 1 x N 2), where N 1 and N 2 are nilpotent Lie groups, and where the action on R contracts the metric on N 1 while extending it on N 2. We obtain new quasi-isometric invariants and classi cations for these spaces.