Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

We study a family of closed connected orientable 3-manifolds obtained by Dehn surgeries with rational coefficients along the oriented components of certain links. This family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge knots. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. As a consequence, we prove that the surgery manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are 2-fold coverings of the 3-sphere branched over well-specified links.


Manifolds Obtained by Dehn Surgeries
As well known, any closed connected orientable 3-manifold can be obtained by Dehn surgeries on the components of an oriented link in the 3-sphere (see [1,2]).If such a link is hyperbolic, then the Thurston-Jorgensen theory [3] of hyperbolic surgery implies that the resulting manifolds are hyperbolic for almost all surgery coefficients.Another method for studying a closed orientable 3-manifold is to represent it as a branched covering of a link in the 3-sphere (see, e.g., [4]).If such a link is hyperbolic, then the construction yields hyperbolic manifolds for branching indices sufficiently large.In the context of current research in 3-manifold topology, many classes of closed orientable hyperbolic 3-manifolds have been constructed by considering branched coverings of links or by performing Dehn surgery along them (see, e.g., [5][6][7][8][9][10]).This paper relates these methods to study a new class of hyperbolic orientable 3-manifolds via combinatorial tools.More precisely, for any positive integer , let L 2+1 be the oriented link with 2 + 1 components  0 ,   , and   ,  = 1, . . ., , in the oriented 3-sphere S 3 depicted in Figure 1.This link can be obtained as a belted sum of Borromean rings, as remarked in [11, p. 8]; thus, it is hyperbolic for any  ≥ 1.Let us consider the closed connected orientable 3-manifolds   (  /  ;   /  ; ℎ/) obtained by Dehn surgery on S 3 along the oriented link L 2+1 such that the surgery coefficients   /  ,   /  , and ℎ/ correspond to the oriented components   ,   , and  0 , respectively, where  = 1, . . ., .Of course, we always assume that gcd(  ,   ) = 1, gcd(  ,   ) = 1, and gcd(ℎ, ) = 1.
Here we will show that our family of manifolds contains all closed manifolds obtained by Dehn surgeries on 2-bridge knots.Such manifolds and their geometries were studied in a nice paper of Brittenham and Wu, where the exceptional Dehn surgeries on 2-bridge knots were completely classified (see [5]).This fact gives a further motivation for the study of our surgery manifolds.Recall that a nontrivial Dehn surgery on a hyperbolic knot in the oriented 3-sphere is said to be exceptional if the resulting manifold is either reducible, toroidal, or a Seifert fibered manifold whose orbifold base is the 2-sphere with at most three exceptional fibers (called a small Seifert fibered space).Thus an exceptional Dehn surgery is not hyperbolic.Moreover, it can be shown that a nonexceptional surgery on a 2-bridge knot is hyperbolic (see [5]).Now we determine a geometric presentation for the fundamental group of the surgery manifold   (  /  ;   /  ; ℎ/).A group presentation is said to be geometric if it arises from a Heegaard diagram of a closed connected (orientable) 3-manifold.If so, then the presentation also corresponds to a spine of the Geometry considered manifold.A Wirtinger presentation of the link group (L 2+1 ) =  1 (S 3 \ L 2+1 ) has generators  0 ,   , and   , for every  = 1, . . .,  (see Figure 1).The meridians m  and   and the longitudes ℓ  and   of the components   and   , respectively, of L 2+1 are where [m  , ℓ  ] = 1 and [  ,   ] = 1 for every  = 1, . . ., .
The meridian m 0 and the longitude ℓ 0 of the component  0 of L 2+1 are To determine the formulae for longitudes ℓ  ,   , and ℓ 0 , we have used the following procedure.Fix an orientation and an initial point for each component of the link L 2+1 .Starting from the initial point, we run along the component in the sense of the fixed orientation and write in order only the generators encountered at the undercrossings.At each undercrossing we write the generator (represented by the oriented arc running over the undercrossing) with positive (resp., negative) exponent if the sense of percorrence is equal (resp., opposite) to the orientation of the named arc.The obtained longitude is homologous to zero in the complement of the considered component if the exponent sum is equal to zero.
Proof.Substituting the above relations in the relators of the Wirtinger presentation of (L 2+1 ) and using the previous formulae for the longitudes ℓ  ,   , and ℓ 0 , we get the relations of the statement.More precisely, substituting or, equivalently, which is the first relation of the statement for  = 1.Then we have or, equivalently, which is the second relation of the statement for  = 1.From the expression of ℓ 2 we get or, equivalently, which is the first relation of the statement for  = 2. From the expression of  2 we get or, equivalently, which is the second relation of the statement for  = 2. Going on like this, we get by finite iteration the first and second relations of the statement for  = 1, . . ., .Substituting ℓ 0 =  −ℎ ,   =     , and we get which gives the last relation of the statement.To show that the presentation in Theorem 1 is geometric, it suffices to draw a suitable RR-system (Rail-Road system) which induces precisely the above presentation (see Figure 2).The hexagons represent the generators, and the three curves labelled by 1, 2, or 3 arrows correspond to the relations in the statement of Theorem 1.For the theory of RR-systems we refer the reader to [12,13].
We also note that the first integral homology group of For example, if   =   = ℎ = 0,  = 1, . . ., , then the Heegaard genus of our surgery manifolds is exactly 2 + 1.
As remarked in [11, p. 8], the link L 2+1 is hyperbolic in the sense that it has a hyperbolic complement.So the Thurston-Jorgensen theory [3] of hyperbolic surgery gives the following result.Theorem 2. For any integer  ≥ 1 and for almost all pairs of surgery coefficients   /  ,   /  , and ℎ/, the closed connected orientable 3-manifolds   (  /  ;   /  ; ℎ/) are hyperbolic.
Since every 2-bridge knot admits a Conway representation with an even number of even parameters (see exercise Geometry Figure 2: An RR-system of genus 2 + 1 inducing the presentation of  1 (  (  /  ;   /  ; ℎ/)).[5]).Since the surgery on torus knot is well understood (see [9]), we restrict our attention to hyperbolic 2-bridge knots.
Ochiai proved that such manifolds have Heegaard genus 2 (see [10]).The following also gives a different proof of the Ochiai result together with an explicit 2-generator 2-relator geometric presentation of the fundamental group.Then the fundamental group of  / () admits a geometric presentation with generators  1 and  and two relators deduced from the recurrence formulae: 1   .In particular, the surgery manifold  / () has Heegaard genus 2.
Proof.By Theorem 1, the fundamental group of  / () has a presentation with generators   ,   , and ,  = 1, . . ., , and relations This presentation is geometric; that is, it is induced by a genus 2 + 1 Heegaard diagram of  / ().We can eliminate the generator 1 to get a balanced presentation of  1 ( / ()) with 2 generators.We see that the curve of the diagram represented by the relator 1 has exactly one point in common with the curve (on the Heegaard surface) represented by the generator  1 .Then the pair of such curves determines a reducible handle in the diagram.Cancelling it yields a new Heegaard diagram of  / () (with genus 2) inducing the above 2-balanced presentation for  1 ( / ()).The recurrence formulae of the statement are obtained as follows: for  = 1, . . .,  − 1.Using these relations we can successively eliminate the generators  +1 and  +1 for  = 1, . . .,  − 1 1   ).The Tietze moves on the obtained presentations for the group  1 ( / ()) correspond geometrically to cancel reducible handles in the current Heegaard diagrams (of decreasing genus) inducing those presentations.So  / () can be represented by a Heegaard diagram of genus 2. Such a diagram induces a geometric presentation for  1 ( / ()) with two generators  1 and  and two relators obtained by applying the above recurrence algorithm.This shows that the genus of  / () is at most 2. Now we claim that the genus is exactly 2. This follows from the fact that 2-bridge knots have tunnel number equal to one and no lens space surgeries (see, e.g., [5]).
Proof.As done in [16, p. 725], for a slightly different link (see also [11,17]), it follows that the links L 2+1 are hyperbolic with volume approximately (2 − 1)(7.32772 . ..).Furthermore, L 2+1 is amphicheiral and its symmetry group is isomorphic to Z 2 ×  4 , where  4 is the dihedral group of order 8. On choosing a framing for each unknotted component of L 2+1 , we can perform 1/ Dehn surgery on each of the unknotted components of L 2+1 .This produces the hyperbolic 2-bridge knot K  =  / , where / = [−2, 2, . . ., −2, 2].Thurston's hyperbolic Dehn surgery theorem [3] in this context says that K  has a 2-long continued fraction consisting of 2's with volumes of S 3 \ K  converging to that of S 3 \ L 2+1 as  goes to infinity.Since these are getting arbitrarily large, the result follows.In fact, the volumes of the surgery hyperbolic manifolds K  (),  ̸ = ∞ and  ≥ 2, become arbitrarily large as  goes to infinity.The fact that the volumes of these manifolds can be arbitrarily large is also a consequence of work by Lackenby on volumes of hyperbolic alternating links (see [18]).(See, e.g., [19,20] for interesting estimates of volumes for hyperbolic manifolds arising from right-angled Coxeter polyhedra.)

Covering Properties
In this section we study covering properties of our surgery manifolds.Using Montesinos' trick [8], we prove that such manifolds are 2-fold branched covers of a connected sum of lens spaces.Moreover, it follows that a very large subclass of our surgery manifolds are 2-fold coverings of the 3-sphere branched over well-specified clearly depicted links.Finally, we show explicitly what the branched cover looks like for the surgeries on a large class of links including 2-bridge knots as very particular case.Theorem 7. Suppose that   is odd for every  = 1, . . ., .Then the surgery manifold   (  /  ;   /  ; ℎ/) is 2-fold branched covering of the connected sum of  lens spaces ( 1 , 2 1 )# ⋅ ⋅ ⋅ #(  , 2  ).
Proof.As shown in Figure 4(a), there is an orientationpreserving involution  in S 3 which induces an involution with two fixed points (resp., without fixed points) in each component  0 and   (resp.,   ) of L 2+1 , for  = 1, . . ., .Here we will assume  ≥ 2. For  = 1 see [6].Let L  be the link consisting of those components of L 2+1 for which the number of fixed points of  is different from two.Let  : S 3 → S 3 be the 2-fold cyclic branched covering of the 3-sphere S 3 defined by .By Theorem 2 of [8] the manifold obtained by doing surgery on L 2+1 is a 2-fold