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Sector estimates for Kleinian groups. (English) Zbl 1068.20053
The author generalizes theorems of S. Lalley [Acta Math. 163, No. 1/2, 1-55 (1989; Zbl 0701.58021)] and P. J. Nicholls [Mich. Math. J. 30, 273-287 (1983; Zbl 0537.30033)]. He considers a convex cocompact Kleinian group \(\Gamma\) which satisfies the ‘even corners’ condition; this is a geometric condition of a technical and non-essential nature. Let \(p,q\) be points of the hyperbolic space on which \(\Gamma\) acts and let \(B\) be a sector centered at \(p\). Let \(N_\Gamma^B(p,q;X)\) be the number of points of \(\Gamma q\) (counted with multiplicity) which lie in the sector \(B\) within a distance of \(X\) from \(p\). Let \(N_\Gamma(p,q;X)\) be the analogous construct when the sector is the entire hyperbolic space. The author shows, using the methods of ergodic theory and symbolic dynamics, that \(N_\Gamma^B(p,q;X)/N_\Gamma(p,q;X)\) converges to \(\mu_{p,q}(\pi_p(B))\) where \(\mu_{p,q}\) denotes the Patterson-Sullivan measure and \(\pi_p\) is the geodesic projection from \(p\) to the boundary.
The ‘even-corners’ condition is used to construct the symbolic dynamics; it seems unlikely that it is essential for the truth of the theorem. The author remarks at the end of paper that the method can be carried over to certain Kleinian groups which are not convex cocompact.

MSC:
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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