The variety of group actions on all algebraic real hyperbolic spaces
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arXiv
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| Format: | Preprint |
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2026
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| author | Duchesne, Bruno Simon, Christopher-Lloyd |
| author_facet | Duchesne, Bruno Simon, Christopher-Lloyd |
| contents | For a cardinal $κ$, denote by $\mathbf{H}^κ$ the algebraic real hyperbolic space of dimension $κ$. For a topological group $Γ$, we study the set of continuous representations $Γ\to \operatorname{Isom}(\mathbf{H}^κ)$ up to continuous self-representations $\operatorname{Isom}(\mathbf{H}^κ)\to \operatorname{Isom}(\mathbf{H}^κ)$.
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 $\mathbf{H}^n$ 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 of algebraic cross-ratio and prove a GNS-embedding result, enabling us to generalize some rigidity properties of the marked length spectrum.
We will also introduce a notion of abstract cross-ratio, and use it to show that a wide class of groups $Γ$ (characterized by the existence of what we call a $3$-full action on a $\operatorname{CAT}(-1)$-space) admit at most one class of irreducible representations into $\operatorname{Isom}(\mathbf{H}^κ)$ whose boundedness properties are controlled by those of $(X,d)$. We will apply this to topological groups $Γ$ such as the isometry group $\operatorname{Isom}(\mathbf{H}^κ)$ itself, the automorphism group $\operatorname{Aut}(T_ω)$ of the simplicial tree with countably infinite valency, and the automorphism group $\operatorname{PGL}_2(\mathbb{K}, \lvert\cdot \rvert)$ of the projective line over a non-Archimedean field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The variety of group actions on all algebraic real hyperbolic spaces Duchesne, Bruno Simon, Christopher-Lloyd Metric Geometry Algebraic Topology Group Theory Representation Theory 22E43, 22E45, 22E50, 22E66, 22F50, 54H15, 57S20, 57S30, 57M30 For a cardinal $κ$, denote by $\mathbf{H}^κ$ the algebraic real hyperbolic space of dimension $κ$. For a topological group $Γ$, we study the set of continuous representations $Γ\to \operatorname{Isom}(\mathbf{H}^κ)$ up to continuous self-representations $\operatorname{Isom}(\mathbf{H}^κ)\to \operatorname{Isom}(\mathbf{H}^κ)$. 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 $\mathbf{H}^n$ 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 of algebraic cross-ratio and prove a GNS-embedding result, enabling us to generalize some rigidity properties of the marked length spectrum. We will also introduce a notion of abstract cross-ratio, and use it to show that a wide class of groups $Γ$ (characterized by the existence of what we call a $3$-full action on a $\operatorname{CAT}(-1)$-space) admit at most one class of irreducible representations into $\operatorname{Isom}(\mathbf{H}^κ)$ whose boundedness properties are controlled by those of $(X,d)$. We will apply this to topological groups $Γ$ such as the isometry group $\operatorname{Isom}(\mathbf{H}^κ)$ itself, the automorphism group $\operatorname{Aut}(T_ω)$ of the simplicial tree with countably infinite valency, and the automorphism group $\operatorname{PGL}_2(\mathbb{K}, \lvert\cdot \rvert)$ of the projective line over a non-Archimedean field. |
| title | The variety of group actions on all algebraic real hyperbolic spaces |
| topic | Metric Geometry Algebraic Topology Group Theory Representation Theory 22E43, 22E45, 22E50, 22E66, 22F50, 54H15, 57S20, 57S30, 57M30 |
| url | https://arxiv.org/abs/2603.03863 |