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Median geometry for spaces with measured walls and for groups
Indira Chatterji1, Cornelia Druţu2
1Laboratoire Dieudonné, Campus Valrose, 06000 Nice, France.
Mathematische Annalen
|January 26, 2026
Summary
Uniform lattices in products of real hyperbolic spaces act properly discontinuously on median spaces. This research explores compatibility with median geometry and geometric characterizations using quasification methods.
Area of Science:
- Geometric group theory
- Topology
- Metric geometry
Background:
- Understanding the geometric properties of discrete groups acting on spaces is fundamental in geometry and topology.
- Median spaces provide a framework for studying geometric structures with properties analogous to trees.
- Hyperbolic spaces are key examples of spaces with rich geometric and group-theoretic properties.
Purpose of the Study:
- To investigate the action of uniform lattices of isometries on products of real hyperbolic spaces.
- To determine the degree of compatibility between these lattices and median geometry.
- To analyze the geometric characterization of spaces near median spaces.
Main Methods:
- Analysis of a quasification of median geometry.
- Geometric characterization of spaces within a finite Hausdorff distance of a median space.
- Study of uniform lattices acting on products of real hyperbolic spaces.
Main Results:
- Uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on median spaces.
- This action represents the strongest possible compatibility with median geometry for lattices in products of at least two factors.
- Complex hyperbolic metric spaces cannot be at a finite Hausdorff distance from a median space.
Conclusions:
- The study establishes a significant connection between hyperbolic geometry and median geometry.
- The findings highlight the distinct behavior of real versus complex hyperbolic spaces in relation to median geometry.
- The quasification of median geometry offers a powerful tool for geometric analysis.
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