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Register a new userComparison of Limit Sets for the Action of Kleinian Groups in $mathbb{C}P^n$
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We compare different notions of limit sets for the action of Kleinian groups on the $n-$dimensional projective space via the irreducible representation $varrho:PSL(2,mathbb{C})to PSL(n+1,mathbb{C}).$ In particular, we prove that if the Kleinian group is convex-cocompact, the Myrberg and the Kulkarni limit coincide.
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We provide a proof of the (well-known) result that the Poincare exponent of a non-elementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof only uses elementary hyperbolic and fractal geometry.
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set; the Poincare exponent of the Kleinian group; and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that the (hyperbolic) boundedness assumption on $C$ cannot be removed in general.
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In this article we show that Bers simultaneous uniformization as well as the Koebes retrosection theorem are not longer true for discrete groups of projective transformations acting on the complex projective space.
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