DISCRETENESS OF HYPERBOLIC ISOMETRIES BY TEST MAPS

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Let F = R, C or the Hamilton’s quaternions H. Let H^n_F denote the n-dimensional F-hyperbolic space. Let U(n,1;F) be the linear group that acts by the isometries of H^n_F. A subgroup G of U(n,1;F) is called Zariski dense if it does not fix a point on H^n_F ∪ ∂H^n_F and neither it preserves a totally geodesic subspace of H^n_F. We prove that a Zariski dense subgroup G of U(n,1;F) is discrete if for every loxodromic element g ∈ G, the two generator subgroup <f,g> is discrete, where f ∈ U(n,1;F) is a test map not necessarily from G.

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  • Osaka Journal of Mathematics

    Osaka Journal of Mathematics 58 (3), 697-710, 2021-07

    Osaka University and Osaka City University, Departments of Mathematics

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