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The critical exponent, the Hausdorff dimension of the limit set and the convex core entropy of a Kleinian group

In this paper we study the relationship between three numerical invariants associated to a Kleinian group, namely the critical exponent, the Hausdorff dimension of the limit set and the convex core entropy. The Hausdorff dimension of the limit set is naturally bounded below by the critical exponent and above by the convex core entropy. We investigate when these inequalities become strict and when they are equalities.