Free actions on handlebodies

Abstract The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g ⩾1 can be enumerated in terms of sets of generators of G . They correspond to the equivalence classes of generating n -vectors of elements of G , where n =1+( g −1)/| G |, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G , this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G , there is only one equivalence class of actions on the genus g handlebody if g is at least 1+l(G) |G| , where l( G ) is the maximal length of a chain of subgroups of G . There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.

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