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Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaard splitting of X. Provided that the genus of V is at least 2, there are similar choices with X closed and hyperbolic.


Introduction
Any finite group acts (smoothly and) freely preserving orientation on some 3-dimensional handlebody, and the number of inequivalent actions of a fixed G on a fixed genus of handlebody can be arbitrarily large [6]. Consequently, the following imbedding property of such actions may appear surprising at first glance: Theorem 1. Let G be a finite group acting freely and preserving orientation on two handlebodies V 1 and V 2 , not necessarily of the same genus. Then there is a G-equivariant imbedding of V 1 into V 2 .
In fact, this result is almost a triviality, as is the following theorem of which it is a special case: Theorem 2. Let G be a finite group acting freely and preserving orientation on a handlebody V and on a connected 3-manifold X. Then there is a Gequivariant imbedding of V into X.
By a result of D. Cooper and D. D. Long [1], any finite group acts freely on some hyperbolic rational homology 3-sphere. So theorem 2 shows that a free G-action on a handlebody always has an extension to an action on such a 3-manifold. Also, by a result of S. Kojima [4], for any finite G there is a closed hyperbolic 3-manifold whose full isometry group is G, and Kojima's construction actually produces a free action. So there is an extension to a free action on a closed hyperbolic 3-manifold whose full isometry group is G.
One might ask for a more natural kind of extension, to a free G-action on a closed 3-manifold M , for which V is one of the handlebodies in a G-invariant Heegaard splitting of M . Simply by forming the double of V and taking an identical action on the second copy of V , one obtains such an extension with M a connected sum of S 2 ×S 1 's. A better question is whether V is an invariant Heegaard handlebody for a free action on an irreducible 3-manifold. Our main result answers this affirmatively.
Theorem 3. Let G be a finite group acting freely and preserving orientation on a handlebody V . Then the action is the restriction of a free G-action on a closed irreducible 3-manifold M , which has a G-invariant Heegaard splitting with V as one of the handlebodies. One may choose M to be Seifertfibered. Provided that V has genus greater than 1, one may choose M to be hyperbolic. In both cases, there are infinitely many choices of M .
We remark that any orientation-preserving action of a finite group on a closed 3-manifold, free or not, has an invariant Heegaard splitting. For the quotient is a closed orientable 3-orbifold with 1-dimensional (possibly empty) singular set. One may triangulate the quotient so that the singular set is a subcomplex of the 1-skeleton. Then, the preimage of a regular neighborhood of the 1-skeleton is invariant and is one of the handlebodies in a Heegaard splitting.
In the remaining sections of this paper, we prove theorems 2 and 3. In [6], a number of results about free G-actions on handlebodies are obtained using more algebraic methods.

Proof of theorem 2
Recall that two G-actions on spaces X and Y are equivalent if there is a homeomorphism j : X → Y such that h(x) = j −1 (h(j(x))) for all x ∈ X and all h ∈ G. If G acts properly discontinuously and freely on a path-connected space X, then the quotient map X → X/G is a regular covering map, so by the theory of covering spaces the action determines an extension Since we have not specified basepoints, the homomorphism φ is well-defined only up to an inner automorphism of G.
Suppose now that G is finite and acts freely and preserving orientation on a handlebody V . The quotient manifold V /G is orientable and irreducible with nonempty boundary, so π 1 (V /G) is torsionfree. A torsionfree finite extension of a finitely generated free group is free (by [3] any finitely generated virtually free group is the fundamental group of a graph of groups with finite vertex groups, and if the group is torsionfree, the vertex groups must be trivial). So π 1 (V /G) is free, and theorem 5.2 of [2] shows that V /G is a handlebody. In this context, we obtain a simple algebraic criterion for equivalence.
Lemma 4. Suppose that G acts freely and preserving orientation on handlebodies V 1 and V 2 , with quotient handlebodies W 1 and W 2 , determining homomorphisms φ i : π 1 (W i ) → G. The actions are equivalent if and only if there is an isomorphism Ψ : Proof. An equivalence of the actions j : Using well-known constructions of homeomorphisms of W 2 (as for example in [5]), all of Nielsen's [7] generators of the automorphism group of the free group π 1 (W 2 ) can be induced by homeomorphisms, so f may be selected to induce Ψ. The condition that φ 2 • Ψ = φ 1 then shows that f lifts to a homeomorphism of covering spaces j : V 1 → V 2 , and moreover ensures that Now we prove theorem 2. Let W be the quotient handlebody of the action on V , let Y = X/G, and let φ : π 1 (W ) → G and ψ : π 1 (Y ) → G be the homomorphisms determined by the actions.
There is an imbedding k : For we can regard W as a regular neighborhood of a 1-point union K of circles, so that π 1 (K) = π 1 (W ), and construct a map Since K is 1-dimensional, k 0 is homotopic to an imbedding, and since W and Y are orientable, this imbedding extends to an imbedding k of W into Y . Since ψ • k # = φ, the preimage of k(W ) in X is connected, and by lemma 4 the restricted G-action on it is equivalent to the original action on V .
Theorem 2 extends to the case when some elements of G reverse the orientation. The equivariant imbedding exists if and only if the subgroups of elements of G that reverse orientation on V and on X are identical. The proof is affected only at the step when the imbedding of K into Y is extended to an imbedding of W into Y . The equality of the orientationreversing subgroups is precisely the condition needed for the extension to exist.

Proof of theorem 3
If the genus of V is 0, then G is trivial and we take M = S 3 . If the genus of V is 1, so that V = D 2 × S 1 , then (using lemma 4) every free G-action is equivalent to a cyclic rotation in the S 1 -factor. Regarding V as a trivially fibered solid torus in the Hopf fibering of S 3 , the action extends to a free action on S 3 with V an invariant Heegaard splitting (it also extends to free actions on infinitely many lens spaces containing V as a fibered Heegaard torus). So we may assume that the genus of V is greater than 1. The quotient handlebody W = V /G has genus at least 2 (since V and consequently W have negative Euler characteristic).
We first construct the Seifert-fibered extension. As in section 1, there is a homomorphism φ : π 1 (W ) → G that determines the action. Let g be the genus of W , and let n be any positive integer divisible by the orders of all the elements of G. We consider a collection of simple closed curves C 1 , . . . , C g in the boundary ∂W , as shown in figure 1 for the case when g = 3 and Let x 1 , . . . , x g be a standard set of generators of π 1 (W ), where x i is represented by a loop that goes once around the i th handle. In π 1 (W ), C i represents x n i (up to conjugacy), and C ′ i represents x n i (x 1 · · · x g ) −n . Since every element of G has order dividing n, it follows that φ carries each C ′ i to the trivial element of G, so induces a homomorphism ψ : π 1 (Y ) → G. If k : W → Y is the inclusion, then ψ • k # = φ. The covering space M of Y has a free G-action and an invariant Heegaard splitting, one of whose handlebodies is the covering space of W corresponding to the kernel of φ, that is, V .
We will show that Y is Seifert-fibered, from which it follows that M is Seifert-fibered. Choose imbedded loops in the interior of W : L near and parallel to C, and L 1 , . . . , L g near and parallel to loops ℓ i in ∂W with each ℓ i going once around the i th handle, meeting C i in one point. The loop ℓ 1 appears in figure 1. By a standard procedure, as explained for example on pp. 275-278 of [9], whose notation we follow, we may change the attaching curves for the discs by Dehn twists about C and the ℓ i , at the expense of performing Dehn surgery on L and the L i . First we twist n times along C, introducing a 1/n coefficient on L and moving each C ′ i back to C i . Then, n − 1 twists along each ℓ i move C i to a loop C ′′ i in ∂W that looks like C i except it goes only once around the handle. This creates surgery coefficients of −1/(n − 1) on the L i . We may change the attaching homeomorphism of the Heegaard splitting by any homeomorphism of ∂W that extends over W , without changing Y . In particular, we may perform left-hand twists in the meridinal 2-discs of the 2-handles to move the C ′′ i to the ℓ i . This subtracts 1 from the surgery coefficients of the L i , and subtracts g from the coefficient of L, yielding the diagram in figure 2. The ℓ i are the attaching curves for  the discs of a Heegaard description of S 3 , so Y is obtained from S 3 by Dehn surgery using the diagram. We can already see a Seifert fibering, but we will work out the exact Seifert invariants. The complement of a regular neighborhood of L is a solid torus T , for which the L i are fibers of a product fibering. A cross-sectional surface in this fibering contained in a meridian disc of T meets the boundary of a regular neighborhood of each L i in a meridian circle and meets the boundary of a regular neighborhood of L in the negative of the longitude, while the fiber meets them in longitude circles and a meridian circle respectively.
A surgery coefficient a/b means that a solid torus is filled in so that a · m + b · ℓ becomes contractible, where m and ℓ are a meridian-longitude pair for a boundary torus of a regular neighborhood of the link component. A Seifert invariant (α, β) determines a filling in which α · q + β · t becomes contractible, where q is the cross-section and t is the fiber. So the surgery coefficients of our link produce one exceptional fiber with Seifert invariant (n, 1−n) for each L i , and one with Seifert invariants (n, gn−1) for L. In the notation of [8], the unnormalized Seifert invariants of Y are {0; (o 1 , 0); (n, 1− n), . . . , (n, 1 − n), (n, gn − 1)}, where there are g + 1 exceptional orbits. The normalized invariants are {−1; (o 1 , 0); (n, 1), . . . , (n, 1), (n, n − 1)}. Now, we will construct the extension of the G-action on V to a hyperbolic 3-manifold. As before, let n be any positive integer divisible by the orders of all the elements of G. Assume for the time being that g = 2. We take the same curves C 1 and C 2 as in the Seifert-fibered construction, but for C we take the image of the loop C 0 shown in figure 3 under the homeomorphism of W which is a right-hand twist in a meridinal disc in each of the two 1-handles.  As before, we take the images of the C i under the n th power of a Dehn twist about C as the attaching curves, form the closed manifold Y , and use the induced homomorphism ψ to obtain the original G-action as an invariant Heegaard handlebody of a free G-action on a covering space M of Y . Let ℓ 1 , ℓ 2 , L 1 , L 2 and L be as before. We choose the L i to lie closer to the boundary of W than L. Again, change the attaching curves first by the inverse of the n Dehn twists about C, introducing a surgery coefficient of 1/n on L, then by the n − 1 twists about the ℓ i , introducing surgery coefficients −1/(n − 1) on the L i . Applying left-hand twists in the meridian discs of the two 1handles of W , we move the attaching curves to the ℓ i , obtaining the surgery description of Y shown in figure 4. This time, the coefficient of L is still 1/n, because L has algebraic intersection 0 with each of the meridian discs of the handles of Y where the left-hand twists were performed. The complement of this link is a 2-fold covering of the complement of the Whitehead link, so is hyperbolic. By [10], Dehn surgery on the link produces a hyperbolic 3manifold, provided that one avoids finitely many choices for the coefficients of each component. So all but finitely many choices for n yield a hyperbolic 3-manifold for Y , and hence for M .
Finally, to adapt the construction to arbitrary genus, one simply adds more components to C 0 to obtain the g − 1 circles shown in figure 5. In the surgery description for Y , the chain L ∪ L 1 in figure 4 is replaced by a chain of length 2g − 2, in which L 1 , . . . , L g−1 alternate with components from C 0 , and the component L g links the chain as did L 2 in figure 4. The L i have surgery coefficients −1 − 1/(n − 1) as before, and the components coming from C 0 have coefficient 1/n, since each had algebraic intersection 0 with the union of the meridian discs. The link complement is the (2g − 2)fold covering of the Whitehead link complement, so is hyperbolic, and the argument is completed as before.