Discreteness of hyperbolic isometries by test maps. (English) Zbl 1542.20239
Summary: Let \(\mathbb{F}=\mathbb{R},\mathbb{C}\) or the Hamilton’s quaternions \(\mathbb{H}\). Let \(\mathbf{H}_{\mathbb{F}}^n\) denote the \(n\)-dimensional \(\mathbb {F}\)-hyperbolic space. Let \(\mathrm{U}(n,1; \mathbb{F})\) be the linear group that acts by the isometries of \(\mathbf{H}_{\mathbb{F}}^n\). A subgroup \(G\) of \(\mathrm{U}(n,1;\mathbb{F})\) is called Zariski dense if it does not fix a point on \(\mathbf{H}_{\mathbb{F}}^n\cup\partial\mathbf{H}_{\mathbb{F}}^n\) and neither it preserves a totally geodesic subspace of \(\mathbf{H}_{\mathbb{F}}^n\). We prove that a Zariski dense subgroup \(G\) of \(\mathrm{U}(n,1;\mathbb{F})\) is discrete if for every loxodromic element \(g\in G\), the two generator subgroup \(\langle f,g\rangle\) is discrete, where \(f \in\mathrm{U}(n,1;\mathbb{F})\) is a test map not necessarily from \(G\).
MSC:
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 20H25 | Other matrix groups over rings |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
| 15A66 | Clifford algebras, spinors |
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