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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:

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