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Let $G$ be the isometry group of a quasi-geodesic Gromov-hyperbolic space $X$. If $X$ is empty there's not much so say, so assume otherwise. Say that $B\subset G$ is bounded if $Bx$ is bounded for som …

What is true is that $G$ has a compact normal subgroup with Fuchsian quotient. First, let $G$ be the isometry group and $W$ the kernel of the action on the Gromov boundary. Then $W$ is compact (as tru …

I don't know if this is the kind of answer you expect, but: In the hyperbolic space of dimension $n+1$ one naturally gets all $n$-dimensional constant curvature geometries. spheres (points at distanc …

Yes, there exists such closed hyperbolic (= constant curvature $-1$) manifolds with this property, in arbitrary dimension. For $d\ge 1$, let $q_t$ be a quadratic form of rank $d$ with coefficients in …

Question 1: in $\mathrm{Isom}(\mathbf{H}^n)$, the centralizer of any loxodromic element preserves its axis, and hence is contained in a closed subgroup isomorphic to $\mathrm{O}(n-1)\times\mathrm{Iso …

This consists in classifying non-elliptic elements of the Lie group $\mathrm{Isom}(\mathbf{H}^n)\simeq\mathrm{PO}(n,1)$ up to conjugacy and inversion. One can do separately loxodromics and horocyclic …