Orientation-reversing free actions on handlebodies (Q2576919)

The authors 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. Mainly they give a bijection from the equivalence classes of free \(G\)-actions on handlebodies of genus \(1+| G| (n-1)\) to the set of Aut(\(F_n\))-orbits in \(\mathcal{G} _n\times \mathbf{V} _n\), where \(\mathcal{G} _n\) is the collection of \(n\)-element generating sets of \(G\) and \(\mathbf{V} _n=\oplus _{i=1}^n C_2, C_2=\{ -1, 1\}\). This is applied to the description of the sets of free actions in various cases. For abelian groups, the free actions are classified for all genera.