Memoirs of the American Mathematical Society 2005; 116 pp; softcover Volume: 177 ISBN10: 0821837729 ISBN13: 9780821837726 List Price: US$64 Individual Members: US$38.40 Institutional Members: US$51.20 Order Code: MEMO/177/834
 Ahlfors conjectured in 1964 that the limit set of every finitely generated Kleinian group either has Lebesgue measure \(0\) or is the entire \(S^2\). We prove that this conjecture is true for purely loxodromic Kleinian groups which are algebraic limits of geometrically finite groups. What we directly prove is that if a purely loxodromic Kleinian group \(\Gamma\) is an algebraic limit of geometrically finite groups and the limit set \(\Lambda_\Gamma\) is not the entire \(S^2_\infty\), then \(\Gamma\) is topologically (and geometrically) tame, that is, there is a compact 3manifold whose interior is homeomorphic to \({\mathbf H}^3/\Gamma\). The proof uses techniques of hyperbolic geometry considerably and is based on works of Maskit, Thurston, Bonahon, Otal, and Canary. Table of Contents  Preliminaries
 Statements of theorems
 Characteristic compression bodies
 The Masur domain and Ahlfors' conjecture
 Branched covers and geometric limit
 Nonrealizable measured laminations
 Strong convergence of function groups
 Proof of the main theorem
 Bibliography
 Index
