In this paper the authors provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space This summarizes and completes a long line of results by many authors from Patterson s classic 1976 paper to more recent results of Hersonsky and Paulin 2002 2004 2007 The authors consider concrete examples of situations which have not been considered before These include geometrically infinite Kleinian groups geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group and groups acting on infinite dimensional hyperbolic space Moreover in addition to providing much greater generality than any prior work of which the authors are aware the results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place Two results are also contained here which are purely geometric a generalization of a theorem of Bishop and Jones 1997 to Gromov hyperbolic metric spaces and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson Sullivan measure unless the group is quasiconvex cocompact The latter is an application of a Diophantine theorem

Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces PDF

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