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DISCRETENESS OF HYPERBOLIC ISOMETRIES BY TEST MAPS

Gongopadhyay, Krishnendu 大阪大学 DOI:10.18910/83208

2021.07

概要

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

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