2001/09/22 by Darryl McCullough, McCullough, Darryl, Marcus Wanderley +1
Computer Science · Mathematics · #57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:57M60 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0109166
arxiv created 2001/09/22 · openalex publication_date 2001/09/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g 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 solvable groups and for the groups PSL(2,3p) with p prime. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+r(G)|G|, where r(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.