We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridianpair-preserving epimorphisms between 2-bridge link groups.

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Riley "defined" the Heckoid groups for 2-bridge links as Kleinian groups, with nontrivial torsion, generated by two parabolic transformations, and he constructed an infinite family of epimorphisms from 2-bridge link groups onto Heckoid groups. In this paper, we make Riley's definition explicit, and give a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups, generalizing Riley's construction.In honour of J. Hyam Rubinstein and his contribution to mathematics

In this paper and its two sequels, we give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. This paper treats the case when the 2-bridge link is a (2, p)-torus link, where more cases of homotopy arise, and its sequels will treat the remaining cases.

Let u be a cyclic word in a free group F n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v| = |u| and v = φ(u) for some φ ∈ Aut F n }. In this paper, we prove that N(u) is bounded by a polynomial function with respect to |u| under the hypothesis that if two letters x, y with x = y ±1 occur in u, then the total number of occurrences of x ±1 in u is not equal to the total number of occurrences of y ±1 in u. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's algorithm for F n .