Curves generated on surfaces by the gilman-maskit algorithm
Jane Gilman, Vidur Malik · 2007
The Gilman-Maskit algorithm determines whether or not two elements of PSL(2, R ) generate a non-elementary discrete group. Gilman-Keen reinterpreted the algorithm as an unwinding and winding of curves about each other on the quotient surface when the group was discrete but did not contain any elliptic or parabolic elements. Gilman-Keen also found a formula to calculate the number of essential self-intersections of these curves. Here, we examine the behavior of the winding and unwinding of the curves in the general case which can include elliptic and parabolic elements. In the case of elliptic elements, when the group is discrete, elements of finite order are present in the group and this makes the quotient an orbifold. We show that elliptic generators create curves that are self-wound. Thus we give a reinterpretation of the algorithm as winding and unwinding of curves in all cases. The second major result of this work is the extension of the Gilman-Keen formula for essential self-intersections to include the cases of elliptic and/or parabolic elements. This new formula takes self-windings into account.