Equivalence and strong equivalence of actions on handlebodies

Abstract

ABSTRACT. An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on 3-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies.

Orientation-reversing free actions on handlebodies

“…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 the work of Kalliongis and Miller in the 1980s, 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.…”

Equivalence of ℤ4-actions on Handlebodies of Genus g

“…Two actions φ 1 and φ 2 on V g are said to be equivalent if and only if there exists an orientation-preserving homeomorphism h of V g such that φ 2 (x) = h • φ 1 (x) • h −1 for all x ∈ G. From [3], the action of any finite group G on V g corresponds to a collection of graphs of groups. We may assume these particular graphs of groups are in canonical form and satisfy a set of normalized conditions, which can be found in [2].…”

“…The graph of groups (Γ(v), G(v)) in canonical form (see [4] for the case p = 2 and p 2 = 4) determines a handlebody orbifold V (Γ(v), G(v)). The orbifold V (Γ(v), G(v)) is constructed in a similar manner as described in [2]. Note that the quotient of any Z p 2 -action on V g is an orbifold of this type, up to homeomorphism.…”

“…). Furthermore, from [4], we may define an equivalence relation and, using techniques found in [2], prove the following.…”

“…We resort to the following lemma to help count the number of elements in the delta set. The proof is an adaptation from [2].…”