Informative Words and Discreteness

Abstract

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{\"u}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of such words and sequences: the semigroup of so called {\sl good} words of Gehring-Martin, the so called {\sl killer} words of Gabai-Meyerhoff-NThurston, the Farey words of Keen-Series and Minsky, the discreteness-algorithm Fibonacci sequences of Gilman-Jiang and {\sl parabolic dust} words. We survey connections between the families and establish a new connection between good words and Farey words.