Reversing Palindromic Enumeration in Rank-Two Free Groups

Abstract

ABSTRACTThe Gilman–Maskit algorithm for determining the discreteness or non-discreteness of a two-generator subgroup of terminates with a pair of generators that are Farey words [Gilman and Maskit 91]. The Farey words are primitive words that are indexed by rational numbers and infinity. The so-called E-words [Gilman and Keen 11], primitive words with palindromic or palindromic product forms, are also indexed by rational numbers and infinity. We produce a modification of the Gilman–Maskit algorithm so that the stopping generators are E-words and what can be considered a new palindromic enumeration scheme. The original definition of the enumeration scheme can be implemented and run in a machine without any modification. However, every time a recursion calls itself, the state of the previous caller is stored until the recursion stops calling itself. It is often efficient for a recursion to minimize calling itself in order to avoid wasted resources such as time and storage space. The non-recursive formulas for special cases reduce the self-calling of the recursion.