Geometric rigidity of quasi-isometries in horospherical products

Ferragut, Tom

doi:10.2140/agt.2026.26.863

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Geometric rigidity of quasi-isometries in horospherical products

Tom Ferragut

Algebraic & Geometric Topology 26 (2026) 863–954

DOI: 10.2140/agt.2026.26.863

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 a result obtained by Eskin, Fisher and Whyte (2012). Our work covers the case of solvable Lie groups of the form $ℝ ⋉ (N_{1} \times N_{2})$ , 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 classifications for these spaces.

Keywords

metric geometry, hyperbolic spaces, horospherical products, solvable groups, Lie groups, quasi-isometry, rigidity, coarse differentiation.

Mathematical Subject Classification

Primary: 22E25, 51F30, 53C24

References

Publication

Received: 16 December 2022

Revised: 4 February 2025

Accepted: 15 April 2025

Published: 1 April 2026

Authors

Tom Ferragut
University Lyon 1 Villeurbanne France

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Volume 26, issue 3 (2026)