Geometry and dynamics of infinitely generated Kleinian groups: Geometric Schottky groups

Abstract

We introduce and study the class of geometric Schottky groups which includes many infinitely generated groups. We define a natural coding of the limit set and characterise in terms of this coding important subsets of the limit set. Further, we study a certain notion of entropy for the geodesic flow on the quotient manifold. We show that for geometric Schottky groups it is equal to the convex core entropy of the group, which for finitely generated groups agrees with the Poincar'e exponent. Subsequently, we discuss a subclass of the geometric Schottky groups called the P-class. For groups in this class, the geodesic flow is ergodic with respect to the Liouville-Patterson measure and the Liouville-Patterson measure is finite. We present proofs of these facts and deduce some consequences. Lastly, we provide methods of constructing groups in the P-class and then show that an infinite-index normal subgroup of a finitely generated geometric Schottky group cannot belong to the P-class.