The variety of group actions on all algebraic real hyperbolic spaces

For a cardinal $\kappa$, denote by $\mathbf{H}^\kappa$ the algebraic real hyperbolic space of dimension $\kappa$. For a topological group $\Gamma$, we study the set of continuous representations $\Gamma \to \operatorname{Isom}(\mathbf{H}^\kappa)$ up to continuous self-representations $\operatorname{Isom}(\mathbf{H}^\kappa)\to \operatorname{Isom}(\mathbf{H}^\kappa)$. 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 $\Gamma$ (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 $\operatorname{Isom}(\mathbf{H}^\kappa)$ whose boundedness properties are controlled by those of $(X,d)$. We will apply this to topological groups $\Gamma$ such as the isometry group $\operatorname{Isom}(\mathbf{H}^\kappa)$ itself, the automorphism group $\operatorname{Aut}(T_\omega)$ 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.