Orientation-reversing Free Actions on Handlebodies

We examine free orientation-reversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Nielsen equivalence. This is applied to describe the sets of free actions in various cases, including a complete classification for many (and conjecturally all) cases above the minimum genus. For abelian groups, the free actions are classified for all genera. The orientation-preserving free actions of a finite group G on 3-dimensional orientable handlebodies have a close connection with a long-studied concept from group theory, namely Nielsen equivalence of generating sets. The basic result is that the orientation-preserving free actions of G on the handlebody of genus g, up to equivalence, correspond to the Nielsen equivalence classes of n-element generating sets of G, where n = 1 + (g − 1)/|G|. This has been known for a long time; it is implicit in work of J. Kalliongis and A. Miller in the 1980’s, as a direct consequence of theorem 1.3 in their paper [7] (for free actions, the graph of groups will have trivial vertex and edge groups, and the equivalence of graphs of groups defined there is readily seen to be the same as Nielsen equivalence on generating sets of G). As far as we know, the first explicit statement detailing the correspondence appears in [13], which also contains various applications and calculations using it. In this paper, we extend the theory from [13] to free actions that contain orientation-reversing elements, and to free actions on nonorientable handlebodies. The orbits of a certain group action on the collection Gn of n-element generating sets are the Nielsen equivalence classes, and this action extends to an action on a set Gn × Vn, in such a way that the orbits correspond to the equivalence classes of all free G-actions on handlebodies of genus 1+(n−1)|G|. This correspondence is given as theorem 1.1, which is proven in section 4 after presentation of preliminary material on Nielsen equivalence in section 2, and on “uniform homeomorphisms” in section 3. From Date: January 30, 2005. 2000 Mathematics Subject Classification. Primary 57M60; Secondary 20F05.