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Hausdorff dimensions of limit sets. I. (English) Zbl 0744.53030

In the paper under review the limit sets of discrete groups of isometries of a Riemannian symmetric space of rank one and of noncompact type are discussed and some of their properties are pointed out. The author relates the Hausdorff dimension of the limit set to the representation theory of Lie groups. The main result is the following: “Suppose \(\Gamma\) is a geometrically cocompact discrete group of isometries of quaternionic hyperbolic space \(H^ n_ \mathbb{H}\), \(n\geq 2\). If \(\Gamma\) is not a lattice, then the limit set has Hausdorff codimension at least 2. An analogous result holds for the Cayley plane \(H^ 2_ \mathbb{Q}\).” There is no analogue of this result in the real hyperbolic case. More precisely, D. Sullivan [Geometry, Proc. Symp., Utrecht 1980, Lect. Notes Math. 894, 127-144 (1981; Zbl 0486.30035)] has given examples of geometrically cocompact groups acting on \(H^ 3_ \mathbb{R}\) whose limit sets have Hausdorff dimensions arbitrarily close to 2.

MSC:

53C35 Differential geometry of symmetric spaces
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods

Citations:

Zbl 0486.30035
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References:

[1] Anderson, M.: The Dirichlet problem at infinity. J. Differ. Geom.18, 701-721 (1983) · Zbl 0541.53036
[2] Burns, D., Shnider, S.: Spherical hypersurfaces in complex manifolds. Invent. Math.33, 223-246 (1976) · Zbl 0357.32012 · doi:10.1007/BF01404204
[3] Cheng, S.Y., Li, P., Yau, S.-T.: On the upper estimate of the heat kernel of a complete Riemannian manifold. Am. J. Math.103, 1021-1063 (1981) · Zbl 0484.53035 · doi:10.2307/2374257
[4] Eberlein, P., O’Neill, B.: Visibility manifolds. Pac. J. Math.46, 45-109 (1973) · Zbl 0264.53026
[5] Epstein, C., Melrose, R., Mendoza, G.: Resolvent of the Laplacian on strictly pseudoconvex domains. Preprint · Zbl 0758.32010
[6] Federer, H.: Geometric measure theory. Berlin, Heidelberg, New York: Springer 1969 · Zbl 0176.00801
[7] Goldman, W.: Geometric structures on manifolds and varieties of representations. In: Geometry of group representations. Goldman, W., Magid, A. (eds.). Contemp. Math.74, 169-198 (1988)
[8] Goldman, W.: A user’s guide to complex hyperbolic geometry. Notes in evolution
[9] Gromov, M.: Asymptotic geometry of homogeneous spaces. In: Differential geometry on homogeneous spaces. Rend. Semin. Math., Fasc. speciale (1983) · Zbl 0627.53036
[10] Helgason, S.: Groups and geometric analysis. Pure and Applied Mathematics113 (1984) · Zbl 0543.58001
[11] Karpelevic, F.: The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trans. Mosc. Math. Soc.14, 48-185 (1965)
[12] Kostant, B.: On the existence and irreducibility of certain series of representations. Bull. AMS75, 627-642 (1969) · Zbl 0229.22026 · doi:10.1090/S0002-9904-1969-12235-4
[13] Kulkarni, R.: Conformal geometry in higher dimensions I. Bull. AMS81, 736-738 (1975) · Zbl 0304.53011 · doi:10.1090/S0002-9904-1975-13847-X
[14] Kulkarni, R.: On the principle of uniformization. J. Differ. Geom.13, 109-138 (1978) · Zbl 0381.53023
[15] Lohou?, N., Rychener, T.: Die Resolvente von ? auf symmetrischen R?ume vom nichtkompakten Typ. Commun. Math. Helv.57, 445-468 (1982) · Zbl 0505.53022 · doi:10.1007/BF02565869
[16] Maskit, B.: Kleinian groups. Berlin, Heidelberg, New York: Springer 1988 · Zbl 0627.30039
[17] Mitchell, J.: On Carnot-Carath?odory metrics. J. Differ. Geom.21, 35-45 (1985) · Zbl 0554.53023
[18] Mostow, G.: Strong rigidity of locally symmetric spaces. Ann. Math. Stud. 78 (1978) · Zbl 0411.22009
[19] Pansu, P.: Une in?galit? isop?rim?trique sur le groupe de Heisenberg. C. R. Acad. Sci. Paris S?r. I,295, 127-130 (1982) · Zbl 0502.53039
[20] Pansu, P.: M?triques de Carnot-Carath?odory et quasi-isom?tries des espaces sym?triques de rang un. Ann. Math.129, 1-60 (1989) · Zbl 0678.53042 · doi:10.2307/1971484
[21] Pansu, P.: Th?se · Zbl 1288.00032
[22] Patterson, S.: The limit set of a Fuchsian group. Acta Math.136, 241-273 (1976) · Zbl 0336.30005 · doi:10.1007/BF02392046
[23] Phillips, R.S., Sarnak, P.: The Laplacian for domains in hyperbolic space and limit sets of Kl?inian groups. Acta Math.155, 173-241 (1985) · Zbl 0611.30037 · doi:10.1007/BF02392542
[24] Strichartz, R.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal.52, 48-79 (1983) · Zbl 0515.58037 · doi:10.1016/0022-1236(83)90090-3
[25] Strichartz, R.: Sub-Riemannian geometry. J. Differ. Geom.24, 221-263 (1986) · Zbl 0609.53021
[26] Sullivan, D.: The density at infinity of a group of hyperbolic motions. Publ. IHES50 171-202 (1979) · Zbl 0439.30034
[27] Sullivan, D.: Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension 2. Geometry Seminar, Utrecht 1980. In: Lecture Notes in Mathematics (eds.) Looijenga, E., Siersma, D., Takens, F.. Berlin, Heidelberg, New York: Springer 1981, Vol. 894, pp. 127-144
[28] Sullivan, D.: Related aspects of positivity in Riemannian geometry. J. Differ. Geom.25, 327-351 (1987) · Zbl 0615.53029
[29] Sullivan, D.: Related aspects of positivity: ?-potential theory on manifolds, lowest eigenstates, Hausdorff geometry, renormalized Markov processes.... In: Aspects of mathematics and its applications. Barroso, J.A., (ed.), pp. 747-779. Amsterdam: North-Holland 1986
[30] Varopoulos, N.: Analysis on nilpotent groups. J. Funct. Anal.66, 406-431 (1986) · Zbl 0595.22008 · doi:10.1016/0022-1236(86)90066-2
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