Geometry and Rank of Fibered Hyperbolic 3-Manifolds

GEOMETR Y AND RA NK OF FIBE RED HYPER BOLIC 3 -MANIF OLDS IAN BIRINGER 1. Introduction Recall that the rank of a finitely generated gr o up is the minimal n umber of eleme nts needed to gen erate it. In [Whi02], M. Wh ite prov ed that the injectivit y radius of a closed hyperb olic 3-manifold M is b ounded ab ov e b y some function of rank( π 1 ( M )). Building o n a tec hnique that he in tro duced, w e determine the ranks of the fundamental groups o f a large class of h yp erb olic 3-manifolds fib ering o v er the circle. Let Σ g b e the closed orien table surface of gen us g and φ : Σ g → Σ g a homeomorphism. W e can construct a 3-manif o ld M φ , the ma pping torus of φ , as the quotien t space M φ = Σ g × [0 , 1] / ∼ , ( x , 0) ∼ ( φ ( x ) , 1) . Th urston [Thu98] has pro v en that if the map φ : Σ g → Σ g is pseudo-anoso v then M φ can b e giv en a hyperb olic metric. The fundamen tal gro up of M φ is give n by an HNN-extension 1 → π 1 (Σ g ) → π 1 ( M φ ) → Z → 1 . Since rank( π 1 (Σ g )) = 2 g it follow s that rank( π 1 ( M φ )) ≤ 2 g +1. It is no t hard to construct examples where this inequalit y is strict, but it seems lik ely that if the gluing map is complicated enough then equality should hold. As an illustration of this, J. Souto pro v ed in [Sou05 ] that g iv en a pseudo-anosov map φ : Σ g → Σ g , we ha v e for sufficien tly large p o we rs φ n of φ that rank( π 1 ( M φ n )) = 2 g + 1. Our main result is the f ollo wing extension of Souto’s theorem. Theorem 1.1. Given ǫ > 0 and a close d orientable s urfac e Σ g , ther e ar e at most fin i tely many ǫ -thick hyp erb olic 3-manifolds M fib ering over S 1 with fib er Σ g for whic h r ank ( π 1 ( M )) 6 = 2 g + 1 . Recall that the inje ctivity r adius of a hy p erb o lic manifold M , written inj( M ), is defined to b e half the length of a shortest homoto pically essen tial lo o p in M , and M is called ǫ -thick if inj( M ) ≥ ǫ . Results similar to Theorem 1.1 concerning the Heegaard genus of M are already kno wn; the strongest is due to Bachm an and Sc hleimer, [BS05]. Recall that the He e gaar d genus of a closed 3-manifold M is the smallest g = g ( M ) 1 2 IAN BIRINGER suc h that M can b e obtained b y gluing tw o genus g handleb o dies alo ng their b oundaries. It is easy to see that when M fib ers o v er the circle with fib er Σ g then g ( M ) ≤ 2 g + 1, and Bachman and Sc hleimer sho w that g ( M ) = 2 g + 1 as long as the mono drom y map of M has translation distance at least 2 g + 1 in the curv e complex of Σ g . It is like ly that the conclusion of Theorem 1.1 is true under similar assumptions, but it is not y et clear t o us how to prov e this. Before b eginning the bulk of this pap er, let us sk etc h the idea b ehind the pro of o f Theorem 1 .1. Let M b e a hy p erb olic 3-manifold fib ering ov er the circle with fib er Σ g . F ollowing a techniq ue of White [Whi02], we find a graph X with rank( π 1 ( X )) = rank( π 1 ( M )) and a π 1 -surjectiv e mapping f : X → M whose image has as small length as p ossible. W e show that if M has large diameter it is most efficien t for X to use small edges to fill out the fundamen tal g roup of t he fib er and a long edge to circumna vigate M in the horizon tal direction. The subgraph of X consisting of all small edges then has rank at least 2 g , since it generates a subgroup of π 1 ( M ) isomorphic to π 1 (Σ g ). But X mus t ha v e ev en larger ra nk, so π 1 ( M ) = π 1 ( X ) ≥ 2 g + 1 . The pap er is organized as follo ws. W e b egin in Section 2 by recalling some standard fa cts from the theory of Kleinian groups. In Section 3, w e use a lemma o f Souto and a compactness argumen t to pro vide geometry b ounds for certain cov ers of doubly degenerate h yp erb olic manifolds homeomorphic to Σ g × R . The minimal length graphs men tioned ab ov e are fo rmally in tro duced in Section 4 and Section 5 con tains a pro of of Theorem 1.1. W e finish with an app endix that fleshes out a result due to Souto, [Sou0 6 ], that giv es a conv enien t decomp osition for minimal length π 1 -surjectiv e graphs in closed hyperb olic 3- manifolds. Ac kno wledgemen ts: I thank Justin Malestein, Nathan Bro addus a nd Benson F arb for their helpful commen ts and Juan Souto for many con v ersations, a dvice and insight. 2. Preliminaries Let M b e a h yp erb o lic 3-manifo ld with finitely generated f undamental group. F or the sak e of simplicit y , w e will assume t ha t M has no cusps. A result of P . Scott in 3- manifold top ology states that M admits a c omp act c or e, tha t is a compact submanifold N whose inclusion in to M is a homot o p y equiv alen ce, [Sc73]. The connec ted comp onen ts of M \ N a re called the en d s of M . Mar- den a sk ed in the 19 70s whether M is alwa ys ho meomorphic to the in terior of its compact core; this w as recen tly pro ve n to b e true by Agol [Agol04] and Calegari-Gabai [CG06]. Consequen tly , if E is an end of M then E is home- omorphic to ∂ E × [0 , ∞ ), where ∂ E is the b oundary comp o nen t of N facing E . GEOMETR Y AND RAN K OF FIBERED H YPERBOLIC 3- MANIFOLDS 3 Define the c onvex c or e of M to b e the smallest conv ex submanifold CC( M ) ⊂ M whos e inclusion is a homotop y equiv alence. An end E of M is called c onvex-c o c omp a ct if E ∩ CC( M ) is compact, and de gener ate otherwise. A con v ex-co compact end is geometrically a warped pro duct, where the metric on lev el surfaces of E ∼ = ∂ E × [0 , ∞ ) gro ws exp onen tially with t he distance to the b oundary of the conv ex core. The geometry of degenerate ends is more subtle - w e will limit ourselv es here to a p ertinen t example and forw ard the reader to [Th u78] and [MT98 ] for the general theory . Example 2.1. L et M φ b e the mapping torus of a pseudo-anoso v map φ : Σ g → Σ g . As men tioned in t he in tro duction, π 1 ( M φ ) decomp oses as 1 → π 1 (Σ g ) → π 1 ( M φ ) → Z → 1 . Let N b e the cyclic cov er of M φ corresp onding to the subgroup π 1 (Σ g ). Then N is homeomorphic to Σ g × R , and since it regularly cov ers a closed manifold w e ha ve CC( N ) = N , implying that b oth ends of N a re degenerate. Note that unw rapping a fib er bundle structure for M φ giv es a pro duct structure N ∼ = Σ g × R with fib ers o f b ounded diameter, contrasting with the exp onen tial gro wth of lev el surfaces in a con v ex-co compact end. 2.1. Simplicial H yp erb olic Surfaces. W e record here some facts ab out negativ ely curv ed surfaces in h yp erb olic 3- manifolds. Definition 2.2. Let M b e a h yp erb olic 3-ma nif o ld. A simp l i c ial hyp erb olic surfac e in M is a map f : S → M , where • S is a closed surface equipp ed with a triangulatio n T • f maps each face of T to a totally geo desic t r ia ngle in M • for eac h v ertex v ∈ T the angles b etw een the f -images of the edges adjacen t to v sum to at least 2 π . If f : S → M is a simplicial h yp erb olic surface then w e get a pa th-metric on S by requiring that f preserv es path lengths. The metric is smo oth and h yp erb olic a wa y from the ve rtices of T , at whic h there a r e p o ssible excesses of angle. By the Gauss-Bonnet Theorem, w e hav e v ol( S ) ≤ 2 π | χ ( S ) | . Bounding the diameter of S b y its volume and injectivity radius, w e obtain: Bounded Diameter Lemma (Bo na hon) . Assume f : S → M is an ǫ -thick simplicial hyp erb olic surfac e of genus g . Then diam( S ) ≤ 4 ǫ (2 g − 2) . Mahler’s Compactness Theorem ([BP92], E.1) states that the mo duli space of ǫ -thic k (smo oth) h yp erb olic surfaces of fixed gen us is compact. T ogether with the following Prop osition, this provide s a n um b er of upp er b ounds on the g eometry of ǫ -thic k simplicial h yp erb olic surfaces , alb eit without explicit constan ts. 4 IAN BIRINGER Prop osition 2.1 (Smo oth D ominates Simplicial) . L et S b e a close d s urfac e and d a metric on S that is the pul lb ack metric for some simpl i c i a l hyp erb olic surfac e. Th e n ther e exists a smo oth hyp erb olic metric d hy p on S such that for al l x, y ∈ S 1 C d ( x, y ) ≤ d hy p ( x, y ) , wher e C > 0 dep ends only on the top olo gic al typ e of S . Note that if d is ǫ -thick then d hy p is ǫ C -thick. Pr o of of Pr op osition 2.1. W orking in p olar co ordinates in small neighborho o ds around the singular p oin ts of d , we can explicitly deform d to obtain a smo oth metric d ′ with Gaussian curv ature K ≤ − 1 t ha t is bilipsc hitz t o d with bilip- sc hitz constan t dep ending only on the angles d has around the p oints in its singular lo cus. The argument is v ery similar to the pro of of t he 2 π -Theorem of Gromov and Th urston [BH96], so w e will omit it here. Since the G auss-Bonnet Theorem giv es an upp er b ound f o r the sum of these singular angles, d and d ′ are in fact C -bilipsc hitz for some C dep ending only on the top ological t yp e of S . Define d hy p to b e the h yp erb olic metric in the conformal class of d ′ . The Ahlfors-Sc h w artz Lemma [Ahl73] states that distances measured in d ′ are less than or equal to distances in d hy p ; this pro ve s the desired inequality .  As an application, w e can use Prop osition 2.1 a nd a based ve rsion of Mahler’s Compactness Theorem to sho w: Corollary 2.2 (Short Markings) . Set Γ = π 1 (Σ g ) and fix a gener ating set X ⊂ Γ . Then given ǫ, g > 0 ther e is a c onstant L such that whenever f : S → M is an ǫ -thick simplicial hyp e rb olic surfac e of genus g and p ∈ S , ther e is an isomorphism Φ : Γ → π 1 ( S, p ) such that the image of e ach elem e n t of X c an b e r e p r esente d by a lo op b ase d at p of length le s s than L . 2.2. Algebraic and Geometric Conv ergence. Let Γ b e a finitely gener- ated gr o up and consider a sequence of discrete and fa ithful represen tations ρ i : Γ → PSL(2 , C ). If ( ρ i ) conv erges p oint wise to ρ ∞ : Γ → PSL(2 , C ), we usually sa y that ( ρ i ) is alg ebr aic al ly c onver gent with ρ ∞ as its algebraic limit. Alternativ ely , consider a sequence of subgroups G i ⊂ PSL(2 , C ); if these con- v erge to a subgroup G ⊂ PSL (2 , C ) in the Hausdorff top olo g y on closed subsets of PSL (2 , C ) then w e say tha t G i → G ge ometric al ly. The case where the t w o notions of conv ergence a g ree is useful enough to w arrant additional terminol- ogy . Sp ecifically , if ρ i → ρ ∞ algebraically a nd ρ i (Γ) → ρ ∞ (Γ) geometrically then one sa ys that ρ i → ρ ∞ str ongly. One can in terpret the geometric conv ergence of a sequence of subgroups G i → G ∞ ⊂ PSL(2 , C ) in terms of the quotient manifolds M i = H 3 /G i . If w e fix a basep oint and baseframe ( p, f ) for H 3 , for each i w e can take t he pro jection ( p i , f i ) as a basep oint and baseframe for M i . Then G i → G ∞ GEOMETR Y AND RAN K OF FIBERED H YPERBOLIC 3- MANIFOLDS 5 geometrically if there exist sequences of p ositiv e num b ers ǫ i → 0 and R i → ∞ , and ( 1 + ǫ i )-bilipsc hitz maps φ i : B( p i , R i ) → M ∞ sending ( p i , f i ) to ( p ∞ , f ∞ ). F or future reference, w e will call the maps φ i a sequence of al m ost isometric ma ps coming from geometric conv ergence. Note that using this a s our definition, w e can speak a b out a geometrically con ve rgent sequence of framed h yp erb olic 3-manif o lds, o r ev en based hyperb olic 3 -manifolds if we forget ab o ut the presence of a baseframe. F or a detailed study o f algebraic and g eometric con v ergence, see [MT98] and [BP92]. 3. Shor t Grap hs in Doubl y Dege nera te Σ g × R Assume that M is a h yp erb olic 3-manifo ld without cusps that is homeomor- phic to Σ g × R . Using W aldhausen’s Cob o rdism Theorem [W al68], it is not hard to see that there is an explicit homeomorphism M ∼ = Σ g × R suc h that CC( M ) sits inside M as either • Σ g × [0 , 1], in whic h case M is conv ex co compact • Σ g × [0 , ∞ ), in whic h case M is called singly degenerate • Σ g × R , and then M is called doubly degenerate. W e men t io ned in the in tro duction that Theorem 1.1 is an extension of an earlier theorem of So uto [Sou05]. A k ey ingredien t in Souto’s pro of was the follo wing observ at io n, whic h is a consequenc e of the Co v ering Theorem of Canary and Th urston [Can96]. Lemma 3.1 ([Sou05]) . L et M b e a do ubly de gen er ate hyp erb olic 3 -mani f o ld home om o rp hic to Σ g × R and let Γ ⊂ π 1 ( M ) b e a pr op er sub gr oup of r ank at most 2 g . Then Γ is fr e e, infinite index and c onvex-c o c omp act. T o prov e Theorem 1.1, w e need an impro v ed v ersion of L emma 3.1 that giv es a diameter b ound for the con v ex core of H 3 / Γ in terms of inj( M ) and the length of a set o f lo ops in M generating Γ. Our pro o f will b e a compactness argumen t: we define a top o logy on the set o f we dges of k b ounded length lo o ps in ǫ -thic k do ubly degenerate h yp erb olic 3-manifolds homeomorphic to Σ g × R , sho w that the resulting space is compact and then use con tinuit y t o sho w that there is an upp er b ound for t he corresp onding con v ex core diameters. Definition 3.1. Define G = G ( ǫ, L, k ) to b e the space of pairs ( M , f ), where (1) M is a doubly degenerate ǫ -t hic k h yp erb o lic 3 - manifold homeomorphic to Σ g × R (2) f : ∧ k S 1 → M is an L -lipschitz map fr om the we dge of k circles, endo w ed with some fixed metric. W e sa y that ( M i , f i ) → ( M ∞ , f ∞ ) if 6 IAN BIRINGER (1) ( M i , ⋆ i ) con v erges geometrically to ( M ∞ , ⋆ ∞ ), where ⋆ i is the w edge p oin t o f f i ( ∧ k S 1 ) (2) there is a sequence φ i of almo st isometric maps coming from the geo- metric con v ergence in (1) suc h tha t φ i ◦ f i : ∧ k S 1 → M con v erges p oin twis e to f ∞ : ∧ k S 1 → M ∞ . Prop osition 3.2. G is c omp act. Pr o of. Let ( M i , f i ) b e a seq uence in G and assume that ⋆ i ∈ M i is the wed ge p oin t of f i ( ∧ k S 1 ). F or eac h i , Canary’s Filling Theorem [Can96 ] giv es a simpli- cial hy p erb olic surface in M i with image passing thro ugh ⋆ i . Using the short markings o f these surfaces pro vided b y Corollary 2 .2 w e can construct repre- sen ta tions ρ i : π 1 (Σ g ) → PSL(2 , C ) with H 3 /ρ i (Σ g ) ∼ = M i so that a fixed base p oin t ⋆ ∈ H 3 pro jects to eac h ⋆ i and up to passing to a subse quence, ρ i con- v erges algebraically to some ρ ∞ : π 1 (Σ g ) → PSL(2 , C ) . Since our low er b ound on injectivit y r a dius p ersists through algebraic limits, ρ ∞ ( π 1 (Σ g )) con tains no parab olics. W ork of Th urston and Bonahon 1 then implies that ρ i → ρ ∞ strongly . Set M ∞ = H 3 /ρ ∞ (Σ g ) and let ⋆ ∞ ∈ M ∞ b e the pro jection of ⋆ . Then ( M i , ⋆ i ) con v erges geometrically to ( M ∞ , ⋆ ∞ ). The fundamen tal group of M ∞ is isomorphic to π 1 (Σ g ), so Bonahon’s T ameness Theorem [Bon86] implies that M ∞ ∼ = Σ g × R . Moreov er, it fo llo ws f r o m strong conv ergence that M ∞ is doubly degenerate. W e can construct a map f ∞ : ∧ k S 1 → M ∞ b y applying Arzela-Ascoli’s Theorem to the sequence of maps φ i ◦ f i : ∧ k S 1 → M ∞ , where φ i is a sequenc e of almost isometric maps coming from g eometric con v ergence. Clearly ( M i , f i ) conv erges to ( M ∞ , f ∞ ) in G .  Corollary 3.3. L et M b e a d oubly de gener ate ǫ -thick hyp erb olic 3 -manifo l d home om o rp hic to Σ g × R and let p ∈ M b e a b asep oint. Assume that Γ ⊂ π 1 ( M , p ) is a pr op er sub gr oup that c an b e gener ate d by 2 g lo ops b ase d at p of length less than L . Then Γ is c onvex c o c om p act and the d i a meter of the c on v ex c or e of H 3 / Γ is b ounde d ab ove by some c onstant dep ending only on L, ǫ and g . Pr o of. Observ e t hat Γ determines a n elemen t ( M , f ) ∈ G = G ( ǫ, L, 2 g ), with the extra prop erty that f is no t π 1 -surjectiv e. The subset of G consisting o f pairs ( M , f ) for whic h f is no t π 1 -surjectiv e is closed in G , and therefore com- pact b y Prop osition 3.2. Lemma 3.1 implies that for all suc h ( M , f ) the cov er M π 1 f of M corresp onding to the π 1 -image of f is con v ex co compact. It is not hard to see that if ( M i , f i ) → ( M ∞ , f ∞ ) ∈ G then ( M i ) π 1 f i → ( M ∞ ) π 1 f ∞ alge- braically after pic king appropriate markings. The diameter of the conv ex core 1 Strong conv ergence follo ws here from tracing throug h Th urston’s pro of of ([Thu78], 9.2) w ith the hindsight provided by Bona hon’s T ameness Theor em [Bon86]. A statement of the resulting theo rem is g iven by Cana ry in ([Can96], 9.1 ) as a prelude to a ser ies of more general co nv e rgence theorems. GEOMETR Y AND RAN K OF FIBERED H YPERBOLIC 3- MANIFOLDS 7 of a h yp erb o lic 3- manifold is contin uous with resp ect to alg ebraic con v ergence, so the diameter o f the conv ex core of ( M i ) π 1 f i v aries con tin uously o v er G . This pro v es the claim.  4. Carrier Graphs In the following, assume M is a closed h yp erb olic 3-manif o ld. Definition 4.1. A c arrier gr aph for M is a graph X and a map f : X → M whic h induces a surjection on fundamen tal g r o ups. Standing Assumption: In this pap er w e are in terested in generating sets of min- imal size, whic h corresp ond to carrier gra phs with rank( π 1 ( X )) = rank( π 1 ( M )). F rom now on all carr ier graphs will b e assumed to hav e this prop erty . If a carrier graph f : X → M is rectifiable, w e can pull bac k path lengths in M to obtain an pseudo-metric o n X . Collapsin g to a p oint eac h zero- length segmen t in X yields a new carrier graph with an actual metric; from no w on we will assume all carrier graphs a re similarly endo w ed. Define the length of a carrier graph to b e the sum of the lengths of its edges, and a minimal length c arrier gr aph to b e a carrier gr a ph whic h has smallest length (o v er all carrier graphs of minimal rank). An argumen t using Arzela-Ascoli’s Theorem, [Whi02], shows that minimal length carrier graphs exist in any closed h yp erb olic 3-ma nif o ld. The following Prop osition sho ws that minimal length carrier graphs a r e geometrically well b ehav ed. Prop osition 4.1 (White, [Whi02 ]) . Assume f : X → M is a minimal length c arrier gr aph in a close d hyp e rb olic 3 -manifold M . Then X is trivalent with 2(rank( π 1 ( M )) − 1) vertic es and 3(rank( π 1 ( M )) − 1) e dges, e ach e dge in X maps to a ge o desic se gment in M , the angle b etwe en any two adjac e n t e dges is 2 π 3 , and the image of any simple close d p a th in X is an essential lo op in M . W e conclude this section with a tec hnical result that is instrumen tal in our pro of of Theorem 1.1. A sligh tly more general theorem w as pro v en b y Souto in [Sou06], but the pro of giv en there is somewhat incomplete. W e include a full pro of of the more g eneral result in App endix A. Prop osition 4.2 (Chains o f Bounded Length) . L et M b e a close d hyp erb olic 3-manifold w ith f : X → M a minimal le n gth c a rrier gr aph. Then we hav e a se quenc e of (p ossibly disc onne cte d) s ub gr aphs ∅ = Y 0 ⊂ Y 1 ⊂ . . . ⊂ Y k = X such that the length of any e dge in Y i +1 \ Y i is b ounde d ab ove by some c onstant dep endin g only on inj( M ) , rank( π 1 ( M )) , length( Y i ) and the diameters of the 8 IAN BIRINGER c onvex c or es of the c overs of M c orr esp onding to f ∗ ( π 1 ( Y j i )) , w h er e Y 1 i , . . . , Y n i ar e the c onne cte d c om p onents of Y i . 5. Proof of Theorem 1.1 Fix ǫ, g > 0 and assume that M is an ǫ -thick hy p erb olic 3-manifold fib ering o v er the circle with fib er Σ g . The g oal of this section is to pro v e that there are o nly finitely man y suc h M for whic h rank( π 1 ( M )) 6 = 2 g + 1. W e b egin, ho w ev er, with a quic k computation concerning M ’s girth. Definition 5.1. The w aist len g th of M , denoted w aist( M ), is the smallest length of a lo o p in M that pro jects to a generator of π 1 ( S 1 ). Prop osition 5.1 (F ib ered 3-Manifolds Hav e Hig h BMI) . L et M b e an ǫ -thick hyp erb ol i c 3 -manifold fib ering over the cir c l e with fib er Σ g . Then 2 diam( M ) − 16 ǫ (2 g − 2) ≤ waist( M ) ≤ 2 diam( M ) . Pr o of. Assume that γ is a lo op realizing the w aist length of M . Canary’s Fill- ing Theorem [Can96] implies that ev ery p oint in the cyclic cov er of M corre- sp onding to the fundamen tal group of the fib er lies in the image of a simplicial h yp erb olic surfaces for whic h t he inclusion map is a homotop y equiv alence. Pro jecting down, this prov ides an exhaustion of M by simplicial h yp erb olic surfaces in the homotop y class of the fib er. By ho mological considerations, an y suc h surface m ust in tersect γ . The Bounded D iameter L emma (see Section 2) then implies that diam( M ) ≤ 1 2 w aist( M ) + 8 ǫ (2 − 2 g ). This establishes the first inequalit y . F or the second, recall that the fundamen ta l group of M is generated b y the set of all lo ops in M of length less than 2 diam( M ) . Any generating set for π 1 ( M ) m ust con tain a lo op that encircles M ’s w aist, so the w aist length of M is at mo st tw ice its diameter.  There are only finitely man y h yp erb o lic 3- manifolds with diameter less than a giv en constan t. Prop osition 5.1 then g ives a similar finiteness result for thic k h yp erb olic 3-manifolds fib ering ov er the circle with a fixed fib er a nd b ounded w aist length. W e are now ready t o prov e the main result of this note. Theorem 1.1. Give n ǫ, g > 0 ther e ar e at most finitely man y ǫ -thick hyp erb olic 3-manifolds M fib ering over S 1 with fib er Σ g for which r ank ( π 1 ( M )) 6 = 2 g + 1 . Pr o of. Assume that M is an ǫ -thic k h yp erb o lic 3- manifold fib ering o ve r the circle with fib er Σ g and rank( π 1 ( M )) ≤ 2 g . W e will sho w that the w aist length of M is b ounded b y some constant dep ending only on ǫ and g . GEOMETR Y AND RAN K OF FIBERED H YPERBOLIC 3- MANIFOLDS 9 Let f : X → M b e a minimal length carrier graph. By Prop osition 4 .2, there is a constant L and a chain of (p ossibly disconnected) subgraphs ∅ = Y 0 ⊂ Y 1 ⊂ . . . ⊂ Y k = X with length( Y i +1 ) b ounded ab o v e by some constan t dep ending only on ǫ , g , length( Y i ) and the dia meters of the con v ex cores of the cov ers of M corre- sp onding to the fundamental groups of the connected comp onen ts of Y i . Assume for the momen t that no connected comp onent of Y i runs all the w a y around M ’s w aist, so that eac h lif ts homeomorphically t o the cyclic co v er M π 1 (Σ g ) of M . Since rank( π 1 ( X )) ≤ 2 g , the comp onen ts of Y i ha v e even smaller rank and th us cannot generate the fundamen tal group of M π 1 (Σ g ) . Therefore Corollary 3.3 applies to b ound the diameters of the asso ciated conv ex cores in terms of length ( Y i ), ǫ a nd g . It follo ws that length( Y i +1 ) is also b ounded ab ov e by length( Y i ), ǫ and g . Applying this a rgumen t iteratively , w e obtain a length b ound for the first subgraph Y i that has a comp onen t whic h na vigates the w aistline of M . The length b ound dep ends on ǫ , g and t he index of the subgraph, but since t here are at most 3(rank( π 1 ( M )) − 1) edges in X the num b er of subgraphs in our c hain is also limited. Therefore we ha v e that the w aist length of M is b ounded b y a function of ǫ and g .  Under sligh t mo difications, the pro o f of Theorem 1.1 show s that for map- ping tori with large w aist length there is only one Nielsen equiv alence class of minimal size generating sets for π 1 ( M ). The intere sted reader ma y compare our pro o f with [Sou06 ] for more details. Appendix A. Chains of Bounded Length W e pro v e here the generalization of Prop osition 4.2 promised in Section 4. The idea o f the pro of given b elo w was originally sk etche d by Souto in [Sou06]; the purp ose of this App endix is t o fill in some missing details. Assume tha t M = H 3 / Γ is a closed hyperb olic 3-manifold and f : X → M is a minimal length carrier gra ph. Cho o se an edge e ⊂ X and a subgraph Y ⊂ X . Our first goal will b e to provide a useful definition o f the length of e relativ e to the subgraph Y . This should v a nish when e ⊂ Y and should ag ree with the hy p erb olic length of f ( e ) when neither of the ve rtices of e lies inside Y . If X is em b edded as a subset of M with f t he inclusion map, then relativ e length is similar to the length e has outside of the h yp erb olic conv ex h ulls of the comp onen ts of Y that e touche s, but w e need to do our measuremen ts in the univ ersal cov er and throw out sections of e that lie inside some of the thin parts of M . T o clarify this, fix a univ ersal cov ering π X : ˜ X → X and a lift ˜ f : ˜ X → H 3 of f . Assume that a ve rtex v of e lies in a connected comp onent Z v ⊂ Y 10 IAN BIRINGER and c ho ose lifts ˆ e, ˆ Z v ⊂ ˜ X of e and Z that touc h ab ov e v . Let Γ ˜ f ( ˆ Z v ) b e the subgroup of Γ tha t lea v es ˜ f ( ˆ Z v ) inv ariant. Definition A.1 (Thic k Con v ex Hulls) . The thick c onv ex hul l of ˜ f ( ˆ Z v ) , written TCH( ˜ f ( ˆ Z v )), is the smallest conv ex set K con taining ˜ f ( ˆ Z v ) suc h that for ev ery γ ∈ Γ ˜ f ( ˆ Z v ) and x ∈ H 3 \ K , we hav e d( γ ( x ) , x ) ≥ 1. Definition A.2 ( Edge Length Relativ e to a Subgraph) . Define the length of e r elative to Y , denoted length Y ( e ), to be the length of the part of ˜ f ( ˆ e ) that lies outside of TCH( ˜ f ( ˆ Z v )) fo r eac h v ertex v of e contained in Y . It is easy t o see t ha t the relativ e length of e is w ell-defined, indep enden t of the lifts c hosen ab ov e. The definition is a bit less complicated if w e assume that X is em b edded as a subset of M . F or then w e can lift e directly to H 3 along with any connected comp onents of Y tha t e touc hes, and then measure the length of e ’s lift outside of the thic k con ve x h ulls of the lifted subgraphs. In t he pro ofs b elo w, w e will a ssume X to b e embedded in order to remo v e a lev el of notational hinderance. The argumen ts will b e exactly the same in the general case. Although an edge can hav e v ery long absolute length while having short length relativ e to a subgraph Y , we can b ound this difference if w e hav e some con trol o v er the geometry o f the co v ers of M corresp onding to the f undamental groups of the comp o nents of Y . Lemma A.1. Assume that M is a clos e d hyp e rb olic 3 -manifold , f : X → M is a mini m al length c arrier gr aph, Y is a sub gr aph of X and e is an e dge of X \ Y . Then length ( e ) is b ounde d ab ove by a c onstant dep e n ding only on length Y ( e ) , length( Y ) , inj( M ) , rank( π 1 ( M )) and the d iameters of the c on vex c or es of the c ove rs of M c orr esp onding to the c omp onents of Y that e touche s . Pr o of. As men tioned ab o v e, we fo rget ab out f a nd assume that X is em b edded as a subset of M . Supp ose that e shares a v ertex with a connected comp onen t Z ⊂ Y , and let ˜ e, ˜ Z ⊂ H 3 b e lifts that touc h ab o v e that vertex . It suffices to show that the Ha usdorff distance from ˜ Z to TCH( ˜ Z ) is b ounded b y the quan tities men tioned in the stat ement of the L emma. F or since X is minimal length, ˜ e ∩ TCH( ˜ Z ) m ust minimize the distance fro m ˜ e ∩ ∂ TCH( ˜ Z ) to ˜ Z ; thus a b ound on the Hausdorff distance betw een ˜ Z and TCH( ˜ Z ) limits the length that ˜ e can hav e inside of TCH( ˜ Z ). W e first claim that the h yp erb olic distance from ˜ Z to CH( Λ (Γ ˜ Z )) is b o unded ab ov e by a constan t depending only on inj( M ) and rank( π 1 ( M )). Cho ose a n infinite piecewise geo desic path γ ⊂ ˜ Z tha t pro jects to a simple closed curv e in Y and let g ∈ Γ ˜ Z b e the corresp onding deck transformation. T aking a maximal sequence o f consecutiv e edges of γ that pro ject to distinct edges in M yields GEOMETR Y AND RAN K OF FIBERED H YPERBOLIC 3- MANIFOLDS 11 a subpath γ ′ whose g -tr a nslates co v er γ . Note that the or t hogonal pro j ection of γ ′ to axis( g ) has length equal to the translation distance of g , whic h is at least inj( M ). By Lemma 4.1 , X has 3(rank( π 1 ( M )) − 1) edges; the n um b er of edges in γ ′ can certainly b e no greater tha n this. Th us there is an edge of γ whose orthogonal pro jection to axis( g ) has length at least inj( M ) 3(rank( π 1 ( M )) − 1) . It follo ws from elemen tary h yp erb olic geometry that there is a p oin t on this edge whose distance from axis( g ) is b ounded ab o v e by a constan t dep ending on that length; this prov es the claim. No w ˜ Z and CH(Λ(Γ ˜ Z )) are b oth in v a rian t under the action of Γ ˜ Z with quotien ts of b ounded diameter, so our limit on the h yp erb olic distance b e- t w een them tra nslates into a b ound on their Hausdorff distance. But if ˜ Z is Hausdorff-close to a con v ex set then it must also b e Hausdorff-close to its con v ex h ull, CH ( ˜ Z ). Since the Hausdorff distance fro m CH( ˜ Z ) to TCH( ˜ Z ) is con trolled b y inj ( M ), we ha ve a b o und on the Hausdorff distance b etw een ˜ Z and TCH( ˜ Z ) .  F or a subgraph Z ⊂ X , w e define the length of Z r elative to Y to b e length Y ( Z ) = X edges e ⊂ Z length Y ( e ) . Using our definition of relative length, w e can streamline the formulation of Prop osition 4.2. The statemen t giv en earlier follows from this one after applying Lemma A.1. Prop osition A.2 (Chains of Bo unded Length) . Ther e i s a universal c ons tant L with the pr op erty that if M is a clos e d hyp erb olic 3-man ifold and f : X → M is a minimal length c arrier gr ap h then we have a se quenc e of (p ossibly disc onne cte d) sub g r aphs ∅ = Y 0 ⊂ Y 1 ⊂ . . . ⊂ Y k = X such that length Y i ( Y i +1 ) < L fo r al l i . Pr o of. It is a standard fact in h yp erb o lic geometry tha t there exist a univ ersal constan t C > 0 with t he following prop ert y: (1) an y path in H 3 made of geo desic segmen ts of length at least C con- nected with angles at least π 3 is a quasi-geo desic. There is also a constan t D > C suc h that (2) if N ⊂ H 3 con tains the axis of a h yp erb olic isometry γ and d( x, γ ( x )) ≥ 1 for a ll x ∈ H 3 \ N , then d( x, γ ( x )) ≥ C for all x ∈ H 3 \ N D ( N ), (3) an y geo desic ray emanating from a con v ex subse t K ⊂ H 3 that leav es N D ( K ) meets ∂ N D ( K ) in an angle of a t least π 3 , and finally a constant B > 0 for whic h 12 IAN BIRINGER (4) an y geo desic exiting a conv ex subset K ⊂ H 3 will exit N D ( K ) after an additional length less than B . W e will sho w that if Y is any subgraph of X then there is an edge in X \ Y of length at most L = C + 2 B relativ e to Y ; applying this iterativ ely will giv e the c hain of subgraphs in the stat ement of the Prop osition. So, supp ose that Y is a subgraph of X . O bserv e that since the fundamen tal group of a closed hy p erb o lic ma nif o ld cannot b e free, there is a n essen tial closed lo op γ ⊂ X that is n ullhomotopic in M . F urthermore, since π 1 ( M ) do es not split as a free pro duct, [Hem76 ], we can pic k γ so that it has no subpath con tained en tirely in Y that is also a closed lo op nullhomotopic in M . Lifting γ to H 3 then give s a closed lo op ˜ γ ⊂ H 3 suc h that each time ˜ γ touc hes a comp onent of π − 1 M ( Y ) it enters and leav es that comp onen t using differen t edges of π − 1 M ( X \ Y ). Consider a maximal segmen t of ˜ γ that is con tained in a comp onen t ˜ Z of π − 1 M ( Y ) and let e and f b e the edges that ˜ γ tra v erses b efor e and after t he segmen t in ˜ Z . If e or f has length less t ha n L relativ e to Y , then we are done. Otherwise, the t w o edges hav e a length of at least L left after exiting TCH( ˜ Z ), so by (4) b oth of these edges m ust exit N D (TCH( ˜ Z )); let e 0 and f 0 b e the p oin ts where they meet ∂ N D (TCH( ˜ Z )). Assume fo r the moment that the distance b etw een e 0 and f 0 is less than C . Then by ( 2 ), e and f pro ject to different edges in X . Substituting π M ( e ∩ N D (TCH( ˜ Z ))) ⊂ X with the pro j ection of the geo desic b et w een e 0 and f 0 therefore yields a new carrier graph for M , and since t he new edge has length less than C while the old has length at least D our new carrier graph has shorter length than X . This con tradicts the minimality of X , so d( e 0 , f 0 ) ≥ C . W e can no w create a new closed path in H 3 from ˜ γ as follows : eac h time ˜ γ tra ve rses a comp onen t ˜ Z of π − 1 M ( Y ), replace the part of ˜ γ that lies inside N D (TCH( ˜ Z )) b y the geo desic with the same endp oints. Then the new path is comp osed of geo desic segmen ts o f length at least C , and by (3), the segmen ts in tersect with angles at least π 3 . Therefore it is a quasi-geo desic. Since it is also closed, this is imp ossible.  Reference s [Agol04] I. Agol. T ameness of hyp erb olic 3-manifolds, preprint (2 004). [Ahl73] Ahlfors, L. Conformal invaria nt s. McGraw-Hill, New Y o rk, 1 973. [CG06] Calegar i, D. & Gabai, D. Shrinkwr apping and the taming of hyp erb olic 3- manifolds. J. Amer. Math. So c. 19 (2006 ), no . 2, 38 5–446 . [BS05] Bachman, D. & Sc hleimer, S. Surfac e bun d les versus H e e gaar d splittings. Comm. Anal. Geom. 13 (2005 ), no . 5, 90 3–928 . [BP92] Benedetti, R. & Petronio, C. L e ctur es on Hyp erb olic Ge ometry. Springer, 19 92. [BH96] Bleiler, S. & Ho dgson, C. Spheric al sp ac e forms and Dehn fil ling. T op ology 35 (1996), no. 3, 809– 833. GEOMETR Y AND RAN K OF FIBERED H YPERBOLIC 3- MANIFOLDS 13 [Bon86] Bonahon, F. Bouts des varietes hyp erb oliques de dimension 3. Ann. of math; 124 (19 86), 71- 158. [Can96] Canary , R. A c overing the or em for hyp erb olic 3-manifolds and its applic ations. T op olog y 3 (1996). [Hem76] Hempel, J. 3 -m anifolds. Princeton Universit y Pres s, 1976 . [MT98] Matsuzaki, K. & T a niguchi, M. Hyp erb olic Manifold s and Kleinian Gr oups. Oxford University P ress, 19 98. [McM96] McMullen, C. R enormalization and 3-manifold s which fib er over the cir cle. An- nals of Mathematics Studies, 142. Princeton Universit y Pres s, Princeton, NJ, 1996. [Sou05] Souto, J. The R ank of t he F undamental Gr oup of Hyp erb olic 3 -manifolds fib ering over the cir cle, preprint. [Sou06] Souto, J . R ank and T op olo gy of Hyp erb olic 3 -manfolds, prepr in t. [Thu 78 ] Thu r ston, W. The ge ometry and top olo gy of 3-manifolds. Lec ture notes, Pr ince- ton University , 1978. [Thu 98 ] Thu r ston, W. Hyp erb olic Stru ctur es on 3 -Manifolds, I: Su rfac e Gr oups and 3 - Manifold s Fib ering over the Cir cle. 1998, preprint. [Sc73] Scott, P . Comp act su bmanifold s of 3 -manifolds. J. London Math. So c., 7, 24 6- 250. [W al68 ] W aldhausen, F. On irr e ducible 3 -manifolds which ar e sufficiently lar ge. Ann. of Math, 87 1968 56 -88. [Whi02] White, M. Inje ctivity r adius and fundamental gr oups of hyp erb olic 3-manifolds. Comm. Anal. Geom. 10 (2 002).