Equivalence and strong equivalence of actions on handlebodies

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. The study of finite group actions on compact 2-manifolds has an intricate history which may be traced back well into the 19th century. (For background on this see the survey article and bibliography [El].) Many of the results which have been established in this study may be obtained through the following algebraic setup: if G is a finite group acting effectively on the compact surface S, then there is an associated group extension 1 ill(S) -+E -+G -+1 obtained by lifting the action to the universal covering of S. Given such a group extension whose abstract kernel G -Out(HI (S)) preserves the peripheral structure of HII(S) (if S is bounded), the affirmative solution to the Nielsen realization problem [K] implies that there is a G-action which corresponds to it. Furthermore it is known that two G-actions on S associated with the same group extension (in the sense of equivalence of exact sequences) are equivalent in fact even strongly equivalent. (In the closed case a proof may be found in [ZZ] for the bounded case a similar approach works using [M].) Within this setting an enumeration of the equivalence classes of G-actions on a given surface is theoretically possible through the techniques employed in [S] and again in [E3]. The key idea is that the set of free G-actions with a fixed quotient surface are algebraically categorized by epimorphisms from the fundamental group of the quotient surface to G. Actions which are not free may be categorized in a similar way using the orbifold fundamental group of the quotient orbifold. In this paper we will utilize a similar approach to study the equivalence and strong equivalence classes of finite group actions on 3-dimensional handlebodies. In Received by the editors April 17, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M99; Secondary 57S25, 57M12, 57M15.