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Median geometry for spaces with measured walls and for groups

Indira Chatterji, Cornelia Druţu

TL;DR

The paper develops a comprehensive link between spaces with measured walls and median spaces, showing that a $\delta$-median structure on a wall space is equivalent to being within finite Hausdorff distance of its associated median space $\mathcal{M}(X)$, and uses this to analyze actions of groups on median spaces. It proves that the real hyperbolic space $\mathbb{H}^n$ embeds into a locally compact median space in a way compatible with its isometry group, yielding proper, cocompact actions of lattices on median spaces, while complex hyperbolic spaces resist such an interpretation due to their snowflake-like distance properties. The work further establishes local compactness of the medianization of real hyperbolic spaces and derives constraints on lattices in products of hyperbolic spaces, showing, for instance, that irreducible lattices with at least two factors cannot act non-trivially on finite-rank median spaces. It also clarifies the limitations for Rips-type theorems in the median setting and highlights precise conditions under which a measured-wall space stays within a bounded distance of its medianization. Overall, the results provide a robust framework for understanding geometric group actions on median spaces and delineate the boundary between real and complex hyperbolic geometries in this context.

Abstract

We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of compatibility possible with the median geometry. Our theorem is also relevant for potential Rips-type theorems for median spaces. The result follows from an analysis of a quasification of median geometry that provides a geometric characterization of spaces at finite Hausdorff distance from a median space. We explain how the case of complex hyperbolic metric spaces is different, and that such spaces cannot be at finite Hausdorff distance from a median space.

Median geometry for spaces with measured walls and for groups

TL;DR

The paper develops a comprehensive link between spaces with measured walls and median spaces, showing that a -median structure on a wall space is equivalent to being within finite Hausdorff distance of its associated median space , and uses this to analyze actions of groups on median spaces. It proves that the real hyperbolic space embeds into a locally compact median space in a way compatible with its isometry group, yielding proper, cocompact actions of lattices on median spaces, while complex hyperbolic spaces resist such an interpretation due to their snowflake-like distance properties. The work further establishes local compactness of the medianization of real hyperbolic spaces and derives constraints on lattices in products of hyperbolic spaces, showing, for instance, that irreducible lattices with at least two factors cannot act non-trivially on finite-rank median spaces. It also clarifies the limitations for Rips-type theorems in the median setting and highlights precise conditions under which a measured-wall space stays within a bounded distance of its medianization. Overall, the results provide a robust framework for understanding geometric group actions on median spaces and delineate the boundary between real and complex hyperbolic geometries in this context.

Abstract

We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of compatibility possible with the median geometry. Our theorem is also relevant for potential Rips-type theorems for median spaces. The result follows from an analysis of a quasification of median geometry that provides a geometric characterization of spaces at finite Hausdorff distance from a median space. We explain how the case of complex hyperbolic metric spaces is different, and that such spaces cannot be at finite Hausdorff distance from a median space.

Paper Structure

This paper contains 6 sections, 22 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Tripodic and median spaces.
  3. The median space associated to a space with measured walls.
  4. Distances to the associated median space.
  5. Local compactness of the medianization of real hyperbolic spaces.
  6. The complex hyperbolic space.

Key Result

Theorem 1.2

Let $(X, \mathcal{W} , \mu )$ be a space with measured walls, $\mu$-locally finite (Definition defn:mulocfinite), and such that all its half-spaces are quasi-convex (Definition quasi-conv). The following are equivalent:

Figures (3)

  • Figure 1: A line $T\times[0,1]$ with pieces of a rectangle removed
  • Figure 2: The Euclidean plane with most walls close to $x$ and crossing $h$, versus the hyperbolic plane where only hyperplanes near $p$ can cross $h$
  • Figure 3: A wall separating $\tau$ from a 1-projection $x$ in $\iota (\mathbb{H}^n)$ also cuts the ball $B(x,\rho)$

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.4
  • Definition 2.5
  • ...and 51 more