The variety of group actions on all algebraic real hyperbolic spaces
Bruno Duchesne, Christopher-Lloyd Simon

TL;DR
This paper explores the space of all continuous group actions on algebraic real hyperbolic spaces of any dimension, establishing topological properties, rigidity results, and classifications for these actions across various groups.
Contribution
It introduces a unified framework for studying all such actions across finite and infinite dimensions, including new compactifications, invariants, and rigidity theorems.
Findings
The set of representations forms a compact topological space.
Recovery of classical compactifications via actions on real trees.
Uniqueness results for certain classes of group actions based on cross-ratio invariants.
Abstract
For a cardinal , denote by the algebraic real hyperbolic space of dimension . For a topological group , we study the set of continuous representations up to continuous self-representations . The novelty of this work relies in considering simultaneously all cardinals, finite or infinite. We will endow this set of classes of representations with a natural topology, and show that this character variety is compact. This will also enable us to recover all previous compactifications of actions on by certain actions on real trees for the equivariant Gromov-Hausdorff topology. A class of representations recovers in particular the homothety class of its marked length spectrum. We will define the notion…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Operator Algebra Research · Mathematical Dynamics and Fractals
