Informative Words and Discreteness

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichmüller 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 good words of Gehring-Martin, the so called killer words of GabaiMeyerhoff-NThurston, the Farey words of Keen-Series and Minsky, the discretenessalgorithm Fibonacci sequences of Gilman-Jiang and parabolic dust words. We survey connections between the families and establish a new connection between good words and Farey words. In 1970’s and 80’s Gerhard Rosenberger and his collaborators developed a theory of discreteness for two generator subgroups of PSL(2,R). Essential to the theory was the concept of replacing pairs of successive generators by Nielsen equivalent pairs in a trace minimizing manner (in order to reach a pair where a discreteness determination could be made). Although they did not use the word algorithm, algorithms and sequences of words were central to their conceptualization of the discreteness problem. In this paper we survey a number of areas where their concepts have had an impact upon the discreteness problem both directly or indirectly. A partial list of the papers they wrote are included in the bibliography.