On The Conical Limit Set Of A Complex Hyperbolic Group

Research Report MRR95-046

On The Conical Limit Set Of A Complex Hyperbolic Group

Gaven Martin and Lesley Ward

Abstract: Let $\Gamma$ be a discrete group of complex hyperbolic isometries of the unit ball of $\Bbb C^n$ . We define and discuss the conical limit set and the exponent of convergence of the Poincaré series for such a group. We show that the exponent of convergence is at most n, and that this bound can be achieved; that the conical limit set has either zero or full Lebesgue measure; and that the conical limit set has measure zero if and only if the Poincaré series converges at the exponent n. These results, while similar to their analogues in real hyperbolic geometry, are not the same. The first of two principal differences between the complex hyperbolic and real hyperbolic settings is that, in the complex hyperbolic case, the cones used to define the conical limit set allow tangential approach in the directions of the complex tangent space. Second, the highly anisotropic nature of the action of a discrete complex hyperbolic group on the boundary of the ball distinguishes this setting from the classical one.

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