Thurston norms of tunnel number-one manifolds
TThurston norms of tunnel number-one manifolds N ATALIA P ACHECO -T ALLAJ K EVIN S CHREVE N ICHOLAS
G. V
LAMIS
The Thurston norm of a 3-manifold measures the complexity of surfaces repre-senting two-dimensional homology classes. We study the possible unit balls ofThurston norms of 3-manifolds M with b ( M ) =
2, and whose fundamental groupsadmit presentations with two generators and one relator. We show that even amongthis special class, there are 3-manifolds such that the unit ball of the Thurston normhas arbitrarily many faces.57M27, 57M05, 20J05, 20J06, 57R19 A knot is the image of a smooth embedding of the circle S into the 3-sphere S . Everyknot K ⊂ S bounds an embedded orientable surface in S − K called a Seifert surface .This leads to a useful knot invariant, called the genus of K , which is the minimal genusof such a Seifert surface. For example, the unknot is the only knot whose genus iszero. This invariant can be generalized to the Thurston norm , which is defined for allcompact, connected, orientable 3–manifolds.Specifically, for each φ ∈ H ( N , Z ), let PD ( φ ) denote the homology class in H ( N , ∂ N , Z ) that is Poincar´e dual to φ . This class can be represented by a properlyembedded oriented surface Σ . Let χ − ( Σ ) = (cid:80) ki = max {− χ ( Σ i ) , } , where Σ , . . . , Σ k are the connected components of Σ . The Thurston norm of φ is defined to be: x N ( φ ) : = min { χ − ( Σ ) | [ Σ ] = PD ( φ ) ∈ H ( N , ∂ N ; Z ) } . Thurston showed that x N can be extended to a seminorm on H ( N , R ). Since a Seifertsurface of a knot K is dual to the generator of H ( S − K , Z ) corresponding to ameridional loop, the Thurston norm of this generator is 2 g ( K ) −
3, where g ( K ) is thegenus of K ; hence, the Thurston norm generalizes the knot genus.The Thurston norm is also useful in studying the possible ways that a 3-manifold canfiber over a circle. Recall that a class φ ∈ H ( N ; R ) is fibered if φ is represented by a a r X i v : . [ m a t h . G T ] M a r Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis non-degenerate closed 1-form; in particular, φ ∈ H ( N , Z ) is fibered if and only if it isinduced by a fibration N → S .The results of Thurston [9, Theorems 1, 2 & 3] can be summarized in the followingtheorem. A marked polytope is a polytope with a distinguished subset of vertices, whichwe refer to as marked vertices . Theorem 1.1 (Thurston)
Let N be a -manifold. There exists a unique centrallysymmetric marked polytope M N in H ( N ; R ) such that for any φ ∈ H ( N ; R ) = Hom( H ( N , R ) , R ) we have x N ( φ ) = max { φ ( p ) − φ ( q ) : p , q ∈ M N } . Moreover, φ is fibered if and only if φ restricted to M N attains its maximum on amarked vertex. The polytope M N is dual to the unit ball of the Thurston norm on H ( N , R ), see Section2. We let P N denote M N without the marked vertices. As a sample application, thisimmediately implies that if a 3-manifold fibers over a circle and the first Betti number b ( N ) is at least 2, then it fibers in infinitely many ways.We are interested in the possible (marked) polytopes that arise as M N for a 3-manifold N . We first note some necessary conditions: If M is the marked polytope of some3-manifold, then(1) M is centrally symmetric, bounded, and convex.(2) M can be translated by a vector v ∈ H ( N , Z ) so that its vertices are in H ( N , Z ) / torsion (this follows from the fact that x N assigns integral values toelements of H ( N , Z )).Thurston showed [9, Corollary to Theorem 6] that for any norm x on R that takes evenintegral values on Z , there is a closed 3-manifold N such that x N is equal to x . Wenote that Thurston’s proof, while explicit, gives no control over the complexity of thefundamental groups of the 3-manifolds constructed.Agol asked the second author whether any polygon in R satisfying (1) and (2) aboveis M N for a very simple 3-manifold N ; in particular, a 3-manifold whose fundamentalgroup admits a (2 , π ( N ) = (cid:104) x , y | r (cid:105) . We cannot givea complete answer, but our main theorem states that the restriction to such simplemanifolds does not restrict the complexity of the associated norms: hurston norms of tunnel number-one manifolds Main Theorem
For any n ∈ N , there exists a 3-manifold N such that π ( N ) admits a (2 , -presentation, M N is a polygon with (2 n ) -sides, every vertex of M N is marked,and M N has (cid:4) n (cid:5) sides of different lengths. The manifolds in the Main Theorem are constructed by gluing a 2-handle onto anorientable genus-two handlebody along a regular neighborhood of a homotopicallynon-trivial simple closed curve (these are usually called tunnel number one manifolds ).Since we require H ( N , R ) ∼ = R , we will always choose the curve to be separating. ByTheorem 1.1, every cohomology class in the examples from the main theorem is fibered.To calculate the Thurston norm of these examples, we use an algorithm of Friedl, thesecond author, and Tillmann [5], which gives an easy way to read off M N from therelator in a (2 , π ( N ). To construct a wide variety of examples, weconsider the orbit of a particular separating curve under the mapping class group of themarked boundary surface of the handlebody, which gives us many possible curves toglue 2-handles onto. Dunfield and D. Thurston have used this same construction witha random walk in the mapping class group to show that a random tunnel number onemanifold does not fiber over a circle [2, Theorem 2.4]. In this sense, our examples arenon-generic.In Section 2 we review the algorithm which computes the marked polytope M N . InSection 3, we recall standard generators of the mapping class group and how they behaveas automorphisms of the fundamental group of a surface. In Section 4 we use thesegenerators to derive a complicated relator r which produces the desired complicatedThurston norm unit ball. In the appendix, we do an explicit calculation of the Thurstonnorm of a 3-manifold with fundamental group the Baumslag-Solitar group BS ( m , m ). Acknowledgements
We would like to thank Nathan Dunfield for providing us with useful Python codeto simplify group presentations, as well as tables of knot group presentations. Thesecond author would like to thank Ian Agol for the problem and Stefan Friedl for usefulsuggestions. This work was completed as part of the REU program at the Universityof Michigan, for the duration of which the first author was supported by NSF grantsDMS-1306992 and DMS-1045119. We would like to thank the organizers of theMichigan REU program for their efforts. The second and third authors were partiallysupported by NSF grant DMS-1045119. This material is based upon work supported bythe National Science Foundation under Award No. 1704364.
Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis
Figure 1: The algorithm applied to the group: (cid:104) x , y | r = xyxyyxyXYXYYXY (cid:105) , where capitalletters denote inverses. This is the first case of our sequence of examples in Section 4. In [5], the following algorithm was given for computing the Thurston norm of 3-manifolds N with π ( N ) = (cid:104) x , y | r (cid:105) and b ( N ) =
2, see also [4]. The relator r determines a walk on H ( N ; Z ) in H ( N ; R ) ∼ = R . We assume that r is reduced andcyclically reduced. The marked polytope M M is constructed as follows (see Figure 1for an example):(1) Start at the origin and walk on Z reading the word r from left to right.(2) Take the convex hull C of the lattice points reached by the walk.(3) Mark the vertices of C which the walk passes through exactly once.(4) Consider the unit squares that are contained in C and touch a vertex of C . Marka midpoint of a square precisely when a vertex of C incident with the square ismarked.(5) The set of vertices of M N is the set of midpoints of these squares, and a vertexof M N is marked precisely when it is a marked midpoint of a square.To obtain the unit ball of the Thurston norm x N , we consider the dual polytope M ∗ N : = { φ ∈ H ( N , R ) : φ ( v ) − φ ( w ) ≤ v , w ∈ M N } . For background and details on mapping class groups, we refer the reader to [3]. Forour purposes, we focus on a single surface: the closed genus two surface Σ . As wewill be working with fundamental groups, we need to keep track of a basepoint. Fix hurston norms of tunnel number-one manifolds a eb dc x z w y γ Figure 2: The right Dehn twists about the curves a , b , c , d , and e generate Mod( Σ ). The loops w , x , y , and z are standard generators for π ( Σ ), which we will use throughout the article. Thecurve γ in π ( Σ ) is the commutator [ x , z ]. a point p in Σ . Let Homeo + ( Σ , p ) denote the group of all orientation-preservingself-homeomorphisms of Σ fixing the point p , and let Homeo ( Σ , p ) be the subgroupof Homeo + ( Σ , p ) consisting of homeomorphisms isotopic to the identity throughisotopies that fix the point p at every stage. Definition 3.1
The mapping class group of the marked surface ( Σ , p ) is the groupMod( Σ , p ) = Homeo + ( Σ , p ) / Homeo ( Σ , p )By keeping track of the point p , we obtain a natural action of Mod( Σ , p ) on π ( Σ , p ).In fact, this action induces an isomorphism from Mod( Σ , p ) to an index-2 subgroup ofAut( π ( Σ , p )) (see [3, Chapter 8]).Given a simple closed curve c on Σ , we let T c ∈ Mod( Σ , p ) denote the isotopy classof a right Dehn twist about c . The five Dehn twists T a , T b , T c , T d , T e about the curves a , b , c , d , e as shown in Figure 2 generate Mod( Σ , p ) (see [3, Theorem 4.13]). In theproof of Theorem 4.3, we will need explicit computations of the action of mappingclasses on elements of the fundamental group; the action of the generators is given inthe lemma below, whose proof we leave as an exercise for the reader. Lemma 3.2
Let w , x , y , z be the standard generators of π ( Σ , p ) shown in Figure 2.If T a , T b , T c , T d , T e are the Dehn twists about the curves a , b , c , d , e as in Figure 2, then • T a : x (cid:55)→ z − x • T b : z (cid:55)→ xz • T c : x (cid:55)→ xwz − , y (cid:55)→ ywz − • T d : w (cid:55)→ y − w • T e : y (cid:55)→ wy • T − a : x (cid:55)→ zx • T − b : z (cid:55)→ x − z • T − c : x (cid:55)→ xzw − , y (cid:55)→ yzw − • T − d : w (cid:55)→ yw • T − e : y (cid:55)→ w − y Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis where an automorphism fixes a generator of π ( Σ , p ) if it is not listed. A handlebody is a compact, orientable, irreducible 3-manifold with a nonemptyconnected boundary whose inclusion is π -surjective. The genus of a handlebody H isdefined to be the genus of ∂ H . We will focus on the genus two handlebody, which onecan visualize as the compact 3-manifold bounded by an embedded copy of a genus twosurface in R (such an embedding is drawn in Figure 2).Given a simple closed curve γ in the boundary of a genus two handlebody H , we definethe 3-manifold M γ to be M γ = H (cid:91) ν ( γ ) ( D × [0 , , where D denotes the 2-disk, ν ( γ ) is a regular neighborhood of γ in ∂ H homeomorphicto γ × [0 , ν ( γ ) is identified with ∂ D × [0 , M γ is the manifoldobtained by gluing a 2-handle onto γ . Lemma 4.1
Let γ be a simple closed curve on the boundary of a genus two handlebody H . If γ is separating (that is, ∂ H (cid:114) γ is disconnected), then b ( M γ ) = , where b denotes the first Betti number. Proof
We can deformation retract M γ to a complex with two 1-cells and one 2-cell.Therefore, b = x and y denote the generators of π ( H ), and x , y , z , w thegenerators of π ( ∂ H ). The attaching map of the two cell comes from removing theletters z and w from the word that γ represents in π ( ∂ H ).If γ is separating, then γ bounds a subsurface of ∂ H , and hence [ γ ] ∈ [ π ( ∂ H ) , π ( ∂ H )].Therefore, the total sum of the exponents of x and y in γ is zero, so the boundary mapof the 2-cell is zero on homology.We now give the construction we will be using in the proof of the Main Theorem. Forthe remainder of the section, we fix H to be a genus two handlebody, p to be a pointin ∂ H , w , x , y , z to be the standard generators of π ( ∂ H , p ) shown in Figure 2, and a , b , c , d , e to be the simple closed curves on ∂ H as shown in Figure 2. In addition, fix asimple, separating closed curve γ on ∂ H representing the commutator [ x , z ] ∈ π ( ∂ H ). hurston norms of tunnel number-one manifolds Definition 4.2
Given an element g ∈ Mod( ∂ H , p ), define M g to be the manifold M g = M g ( γ ) . Theorem 4.3
For each n ∈ N let g n = T − b ( T − d T c ) n + . The polygon M M gn has n marked vertices. Remark
Here are the first two relators that this algorithm constructs (capital lettersdenote inverses). (1) xyxyyxyXYXYYXY (2) xyxyyxyxyyxyyxyxyyxyXYXYYXYXYYXYYXYXYYXYSuccessive relators have a similar structure; namely, the first half of the relator consistsof a word containing only positive powers of x and y with the second half obtainedfrom the first by replacing each instance of x and y with their inverses. Proof of Theorem 4.3
Let g = T − b T − d T c T b , so that g n + = g · g n . We willinductively define two sequences of words in the generators of π ( ∂ H , p ): A = ywz − x A n + = B n A n B = y wz − x B n + = B n B n A n = B n A n + We claim that(4–1) A n + = gA n and B n + = gB n . We proceed by induction. The base case is established by a straightforward computation: gA = B A = A and gB = B B A = B . Now let n ∈ N and assume gA n − = A n and gB n − = B n . We then have that gA n = g ( B n − ) g ( A n − ) = B n A n = A n + and gB n = g ( B n − ) g ( B n − ) g ( A n − ) = B n B n A n = B n + . Given any word A in { w , x , y , z } , let ¯ A denote the word obtained by eliminating theletters w and z , and then reducing and cyclically reducing. The images of x and y generate π ( H , p ), and the word ¯ A determines a walk in H ( M v , Z ) ∼ = Z as in Section2. Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis
Let w ( ¯ A n ) and w ( ¯ B n ) denote the width of these walks and h ( ¯ A n ) and h ( ¯ B n ) their height.We have w ( ¯ A ) = , h ( ¯ A ) = , w ( ¯ B ) = , h ( ¯ B ) = A n and B n contain only positive powers of x and y that w ( ¯ A n + ) = w ( ¯ B n ) + w ( ¯ A n ) h ( ¯ A n + ) = h ( ¯ B n ) + h ( ¯ A n ) w ( ¯ B n + ) = w ( ¯ B n ) + w ( ¯ A n + ) h ( ¯ B n + ) = h ( ¯ B n ) + h ( ¯ A n + ) . Let f n denote the n th term of the Fibonacci sequence starting with f = f =
1. Anothershort induction argument yields(4–2) w ( ¯ A n ) = f n − , h ( ¯ A n ) = f n − , w ( ¯ B n ) = f n − , and h ( ¯ B n ) = f n We claim that(4–3) g n ([ x , z ]) = xA A . . . A n B n B n − . . . B yw ( g n ( x − z − ))This can easily be checked for n = g ( x ) = xA and g ( yw ) = B yw . Fix n ∈ N and define the list of points P n in Z as follows: P n = { (1 , , ( w ( ¯ A ) , h ( ¯ A )) , . . . , ( w ( ¯ A n ) , h ( ¯ A n )) , ( w ( ¯ B n ) , h ( ¯ B n )) , . . . , ( w ( ¯ B ) , h ( ¯ B )) , (0 , } = { (1 , , ( f , f ) , ( f , f ) , . . . , ( f n − , f n − ) , ( f n − , f n ) , ( f n − , f n − ) , . . . , ( f , f ) , (0 , } . Note that | P n | = n +
2. For k ∈ { , , . . . , | P n |} , define the point p k in Z by setting p = (0 ,
0) and p k = p k − + P n ( k )for 0 < k ≤ | P n | , where P n ( k ) denotes the k th element in the list P n . Additionally,we define (cid:96) k to be the line segment connecting consecutive points, that is, for k ∈{ , . . . , | P n |} we set (cid:96) k = p k − p k . hurston norms of tunnel number-one manifolds A B A A g n ( x − z − ) Figure 3: A general picture of g n ([ x , z ]), after removing the letters z and w . The relator starts atthe origin and climbs up the bottom “staircase” which is g n ( xz ). Then g n ( x − z − ) travels backdown. It follows from the construction that for each k ∈ { , . . . , | P n |} the point p k lies on thewalk determined by g n ( xz ) = xA A . . . A n B n B n − . . . B yw . Moreover, for k ∈ { , . . . , n + } , the point p k + is the endpoint of the walk determinedby xA . . . A k . Similarly, for k ∈ { n + , . . . , n + } , the point p k + is the end pointof the walk determined by xA . . . A n B n . . . B n − k + . We now want to prove that thewalk determined by g n ( xz ) is bounded on the right by (cid:83) k (cid:96) k .We again proceed by induction on both sequences: As our base case we observe thatthe vertices in the walk determined by ¯ A are (0 , , (0 ,
1) and (1 , B . Now the walk determined by ¯ A k is the concatenation of the walkdetermined by ¯ B k − followed by that of ¯ A k − . Now it follows from (4–2) that the slopeof the line connecting the endpoints of the walk corresponding to ¯ B k − is f k − f k − and theslope connecting the endpoints of the walk given by ¯ A k is f k − f k − . Standard properties ofthe Fibonacci sequence guarantee the latter is always smaller. If we now assume that allthe vertices in the walks ¯ B k − and ¯ A k − are bounded to the right by the line segmentsconnecting their respective endpoints, then the same is true for the walk given by ¯ A k . A Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis similar argument can be made for ¯ B k . As the walk given by g n ( xz ) is a concatenationof the walks given by the ¯ A k and ¯ B k ’s, we see that (cid:83) k (cid:96) k bounds this walk on the right.Let Θ n : R → R be defined by Θ n ( v ) = − v + p n + . We now claim that the convexhull C n of the walk given by g n ([ x , z ]) has vertex set V n = { p k : 0 ≤ k ≤ n + } ∪ { Θ n ( p k ) : 0 < k < n + } and edge set E n = { (cid:96) k : 1 ≤ k ≤ n + } ∪ { Θ n ( (cid:96) k ) : 1 ≤ k ≤ n + } . Let ˆ (cid:96) k be the complete line containing the segment (cid:96) k . Again using basic properties ofthe Fibonacci sequence, the slope of ˆ (cid:96) k is strictly greater than that of ˆ (cid:96) k + implyingthat each ˆ (cid:96) k bounds C n from the right. Now combining this with the fact that (cid:96) k iscontained in C n , we must have that (cid:96) k is an edge of C n . As the walk is invariant underthe transformation Θ n , we see that Θ n ( (cid:96) k ) is also an edge of C n . Finally, the pointwiseunion of the elements of E n bounds a convex polygon and hence E n contains all theedges of C n .To finish, we note that C n has 4 n + M M gn with 4 n vertices. In par-ticular, the two collections of vertices { Θ n ( p n + ) , p , p } and { p n + , p n + , Θ n ( p ) } each collapse to a single vertex in the process of obtaining M M gn from C n (see Figure4). It is easy to see that two vertices v and v (cid:48) of the convex hull collapse to a singlevertex in M π only if they lie on the same square, so no other vertices of the convexhull collapse.Similar arguments give the following theorem; we only sketch the proof. Theorem 4.4
For each n ∈ N let h n = T − b ( T − d T c ) n + . The polygon M M hn has n + marked vertices. Proof
With g n , A k , B k as in the proof of Theorem 4.3, the composition of T − b with g n simply increases the width of A , B by 1. Therefore, the Fibonacci sequences startat f = f = A k , B k . Exactly the same argument asabove applies to obtain a convex hull of 4 n + n + Remark
We do not know if the -manifolds that we construct have any alternativedescriptions. For example, it would be interesting if these manifolds were all link hurston norms of tunnel number-one manifolds Θ ( p n + ) • Θ ( l n + ) p • l p • l p • l p • p n + • l n + p n + • Θ ( l ) Θ ( p ) • Θ ( l ) Θ ( p ) • Θ ( l ) Θ ( p ) • Figure 4: The marked polytope M M gn with 4 n vertices. complements that had a uniform construction. In [7] , Licata used Heegaard Floerhomology to compute M π for all two-component, four-strand pretzel links, and theunit balls are shaped similarly to the polytopes we construct. On the other hand, all ofLicata’s examples have 8 vertices, so cannot match our examples. We study in detail a specific collection of 3-manifolds that were helpful for us inunderstanding the Thurston norm of tunnel number one manifolds. In particular, welook at 3-manifolds whose fundamental group is isomorphic to a Baumslag-Solitargroup of the form BS(m , m) for some positive integer m , that is, π = (cid:104) x , y | xy m x − y − m (cid:105) . For such a manifold, the Thurston norm assigns 0 to the cohomology class dual to x ∈ H ( M ), and m − y . This can be seen using the algorithmpresented in Section 2. Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis
Figure 5: Shown here is the manifold M f m ∼ = N = M ∪ M with m =
4. The manifold M isdrawn on the left and M on the right. Curves of the same color are identified in M f m . In the notation of Section 4, if for a positive integer m we define f m ∈ Mod( ∂ H ) to be f m = T md T − c T d T c , then π ( M f m ) = BS( m , m ).For the purpose of this appendix, we give another useful construction of the manifold M f m : Let M = S × D be a solid torus, ( p , q ) ∈ ∂ M , λ = S ×{ q } , and µ = { p }× ∂ D .Let γ be a curve in ∂ M representing the homology class m [ λ ] + [ µ ] in H ( ∂ M ) andlet γ and γ be distinct parallel copies of γ , that is, γ, γ , and γ are pairwise disjointand homologous. Now, let M = S × [0 , × [0 , ν ( γ ) and ν ( γ ) be disjoint regular neighborhoods of γ and γ , respectively. Let γ + i and γ − i denote the boundary components of ν ( γ i ) labelled such that γ and γ arecontained in distinct components of ∂ M (cid:114) ( γ + ∪ γ + ). Let N be the manifold obtainedby identifying ν ( γ i ) in ∂ M with the annulus S × [0 , × { i } ⊂ ∂ M such that • γ + i is identified with S × { − i } × { i } and • γ − i is identified with S × { i } × { i } .Up to homotopy, we may assume that γ is contained in the annulus A bounded by γ + and γ − in ∂ M . Pick a point t ∈ S and let δ be the loop obtained by closing upthe segment { t } × { } × [0 ,
1] in ∂ M with an arc contained in A and intersecting γ once. We can then see that γ and δ are contained a torus component of ∂ N andhence commute as elements of the fundamental group. Furthermore, an application ofVan Kampen’s theorem shows that the curves λ and δ generate π ( N ) and yields thepresentation π ( N ) = (cid:104) δ, λ | δλ m δ − λ − m (cid:105) . This construction is shown in Figure 5.Now, it is easy to see that BS( m , m ) has an infinite and normal cyclic subgroup (e.g.take the group generated by λ m in π ( N )). The Seifert fibered space theorem (see [1, 6]) hurston norms of tunnel number-one manifolds ++ Figure 6: The dual disk to λ in the solid torus, and its extension to a dual surface in N . implies that both M f m and N are Seifert fibered. In [8, Theorem 3.1], Scott showed thatif two Seifert fibered spaces have infinite and isomorphic fundamental groups, then theyare in fact homeomorphic; hence, N and M f m are homeomorphic.Using this new description of M f m we can find the minimal dual surfaces to λ and δ . By applying the algorithm in Section 2, we know there exists a surface of Eulercharacteristic 0 dual to δ , e.g. the surface S × [0 , × { } ⊂ M .Let us now focus on λ . In M , λ is dual to the disk D bounded by µ ; however, µ does not live in the boundary of M ; we will fix this with a surgery. For i ∈ { , } ,the intersection ν ( γ i ) ∩ µ can be written as a disjoint union of m intervals, call them I i , . . . , I im (recall that | γ ∩ µ | = m by construction). The intervals I k and I k bound arectangle R k in M of the form I k × [0 ,
1] for each k ∈ { , . . . , m } . Now the surface Σ = D ∪ R ∪ · · · ∪ R m is obtained from a disk by attaching m rectangular strips, so it is a genus-0 surface with m boundary components. In particular, χ ( Σ ) = − m . Furthermore, Σ is dual to λ .The algorithm guarantees this dual surface is minimal. Natalia Pacheco-Tallaj, Kevin Schreve and Nicholas G. Vlamis
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Convergence groups and Seifert fibered -manifolds , Invent. Math. (1994), no. 3, 441–456.[2] N. Dunfield and D. Thurston, A random tunnel number one 3-manifold does not fiber over the circle ,Geometry & Topology (2006), 2431-2499.[3] B. Farb and D. Margalit, Primer on Mapping Class Groups , Princeton University Press, 2012.[4] S. Friedl and S. Tillman,
Two-generator one-relator groups and marked polytopes . preprint.[5] S. Friedl, K. Schreve, and S. Tillman,
Thurston Norm and Fox Calculus , Geometry & Topology (2017), 3759-3784.[6] D. Gabai, Convergence groups are Fuchsian groups , Ann. of Math. (2) (1992), no. 3, 447–510.[7] J. Licata,