Galois conjugates of pseudo-Anosov stretch factors are dense in the complex plane
aa r X i v : . [ m a t h . G T ] A ug GALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCHFACTORS ARE DENSE IN THE COMPLEX PLANE
BAL ´AZS STRENNER
Abstract.
In this paper, we study the Galois conjugates of stretchfactors of pseudo-Anosov elements of the mapping class group of a sur-face. We show that—except in low-complexity cases—these conjugatesare dense in the complex plane. For this, we use Penner’s constructionof pseudo-Anosov mapping classes. As a consequence, we obtain thatin a sense there is no restriction on the location of Galois conjugatesof stretch factors arising from Penner’s construction. This complementsan earlier result of Shin and the author stating that Galois conjugatesof stretch factors arising from Penner’s construction may never lie onthe unit circle. Introduction
Let S be a compact orientable surface. The Nielsen–Thurston classifica-tion theorem [Thu88] states that every element f of the mapping class groupMod( S ) is either finite order, reducible or pseudo-Anosov. Associated to ev-ery pseudo-Anosov element is a stretch factor λ > S g,n be the orientable surface of genus g with n boundary components.We define the complexity of S g,n as ξ ( S g,n ) = 3 g − n . Note that ξ ( S ) ishalf the dimension of the Teichm¨uller space of S .The main result of the paper is the following. Theorem 1.1. If S is a compact orientable surface with ξ ( S ) ≥ , then theGalois conjugates of stretch factors of pseudo-Anosov elements of Mod( S ) are dense in the complex plane. We proceed with providing motivation for the theorem. Then, at the endof the introduction, we give an outline of the proof.
Relation to Fried’s problem.
Every pseudo-Anosov stretch factor is a bi-Perron algebraic unit: an algebraic unit λ > λ and 1 /λ lie in the annulus 1 /λ < | z | < λ . Fried [Fri85] asked whetheror not the converse holds (up to taking powers), and it became a folkloreconjecture that it does. This would give a characterization of the numbersthat arise as pseudo-Anosov stretch factors. Assuming this conjecture, one Date : March 9, 2018. would expect the Galois conjugates of pseudo-Anosov stretch factors to bedense in the complex plane. Theorem 1.1 is consistent with the conjecture.
Related results.
Other than the bi-Perron property, little is known aboutthe Galois conjugates of pseudo-Anosov stretch factors. Nevertheless, stretchfactors of maps appear in many different but related contexts, where someresults about the Galois conjugates are available.Hamenst¨adt [Ham14, Theorem 1] showed that (in an appropriate sense)typical stretch factors of the homological actions of pseudo-Anosov mappingclasses are totally real. If the typical pseudo-Anosov stretch factor was alsototally real, then the pseudo-Anosov stretch factors we construct in thispaper would be atypical, since their Galois conjugates are everywhere in thecomplex plane.Thurston [Thu14] studied the stretch factors of graph maps, outer au-tomorphisms of free groups and post-critically finite self-maps of the unitinterval. He gave a characterization of such stretch factors in terms of thelocation of Galois conjugates: they are the so-called weak Perron numbers.Following up on Thurston’s work, Tiozzo [Tio13] studied the fractal de-fined as the closure of the Galois conjugates of growth rates of superattract-ing real quadratic polynomials and showed that this fractal is path-connectedand locally connected. An analogous fractal for pseudo-Anosov stretch fac-tors would be the closure of the Galois conjugates of pseudo-Anosov stretchfactors λ satisfying λ ≤ T for some T >
1. As far as we know, this fractalhas not yet been studied.
Construction of pseudo-Anosov mapping classes.
To prove Theo-rem 1.1, we use the following construction of pseudo-Anosov mapping classes[Pen88] (see also [Fat92]).
Penner’s Construction.
Let A = { a , . . . , a n } and B = { b , . . . , b m } bea pair of filling multicurves on an orientable surface S . Then any productof positive Dehn twists T a j and negative Dehn twists T − b k is pseudo-Anosovprovided that each curve is used at least once. For each pair of multicurves C = A ∪ B , we denote by GP ( C ) the set ofGalois conjugates of stretch factors of pseudo-Anosov elements of Mod( S )arising from Penner’s construction using the set of curves C . Theorem 1.1is a corollary of the following more concrete statement. Theorem 1.2. If S is a compact orientable surface with ξ ( S ) ≥ , thenthere is a collection of curves C on S such that GP ( C ) = C . Penner [Pen88] asked if every pseudo-Anosov mapping class has a powerthat arises from his construction. This was answered in the negative by Shinand the author [SS15] by showing that stretch factors arising from Penner’sconstruction do not have Galois conjugates on the unit circle. In Question The components of the complement of A and B are disks or once-punctured disks. ALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCH FACTORS 3 C . Theorem 1.2 answersthis question positively. Low complexity cases.
The hypothesis on the complexity of the surfacein Theorem 1.1 is necessary, because when ξ ( S ) ≤
2, the Galois conjugatesof stretch factors lie on the real line and the unit circle. This is for thefollowing reasons.Pseudo-Anosov stretch factors arise as eigenvalues of integral symplecticmatrices of size 2 ξ ( S ) × ξ ( S ). These matrices come from the integral piece-wise linear action of the mapping class on the measured lamination space of S which has dimension 2 ξ ( S ). A symplectic matrix that has an eigenvalueoff the real line and the unit circle has at least 4 such eigenvalues (by com-plex conjugation and the fact that eigenvalues come in reciprocal pairs), soif it also has a positive real eigenvalue, its size has to be at least 6 × ξ ( S ) = 2. However, when restrictingto Penner’s construction, the Galois conjugates can only be positive real.The fact that they cannot lie on the unit circle was mentioned earlier. Thefact that they cannot be negative can be found in the author’s thesis [Str15,Section 6.2]. Sketch of the proof.
We divide the proof into three parts, correspondingto Sections 2 to 4.In Theorem 2.1, we give a sufficient condition for a complex number to becontained in GP ( C ) in terms of the eigenvalues of compositions of certainprojections from hyperplanes to other hyperplanes in R n . This reduces toproblem of approximating complex numbers by Galois conjugates of stretchfactors to approximating complex numbers by eigenvalues of compositionsof projections. The proof of this uses results from [Str16] stating that forcertain sequences of pseudo-Anosov mapping classes arising from Penner’sconstruction, some Galois conjugates of the stretch factors converge, andthe limits are eigenvalues of a composition of projections.In Section 3, we define the notion of rich collections of curves, and in The-orem 3.2 we show that if C is a rich collection of curves, then GP ( C ) = C .The main ingredient to this is showing that if C is a rich collection, thenevery invertible linear transformation of the 2-dimensional plane can be ap-proximated by compositions of certain projections from 2-dimensional planesto other 2-dimensional planes in R . This allows us to apply Theorem 2.1to conclude that all complex numbers are contained in GP ( C ).Finally, in Section 4 we construct rich collections of curves on varioussurfaces and complete the proof of Theorem 1.2. Nonorientable surfaces.
Penner’s construction also works on nonorientablesurfaces [Pen88, Str16], and an analog of Theorem 1.2 could be proven alsofor sufficiently complicated nonorientable surfaces. In the orientable case,
BAL ´AZS STRENNER we deduce Theorem 1.2 as a corollary of Theorem 3.2 and Proposition 4.1.Theorem 3.2 applies to the nonorientable case as it is, so one would onlyneed to construct rich collections of curves on nonorientable surfaces.2.
Galois conjugates of stretch factors in Penner’sconstruction
The goal of this section is to establish a connection between Galois con-jugates of pseudo-Anosov stretch factors and eigenvalues of certain compo-sitions of projections.Let C = { c , . . . , c n } be a collection of curves used in Penner’s construc-tion. The intersection matrix Ω = i ( C, C ) is the n × n matrix whose ( j, k )-entry is the geometric intersection number i ( c j , c k ).Let Z i be the orthogonal complement of the i th row of Ω. Since Ω is anintersection matrix of a collection of filling curves, all rows are nonzero andthe Z i are hyperplanes. Let p i ← j : R n → Z i be the—not necessarily orthogonal—projection onto the hyperplane Z i inthe direction of e j , the j th standard basis vector in R n . This projection isdefined if and only if e j is not contained in Z i , which is in turn equivalentto the statement that the ( i, j )-entry of Ω is positive.Let G (Ω) be the graph on the vertex set { , . . . , n } where i and j areconnected if the ( i, j )-entry of Ω is positive. For a closed path γ = ( i · · · i K i )in G (Ω), define the linear map f γ : Z i → Z i by the formula f γ = ( p i ← i K ◦ · · · ◦ p i ← i ) | Z i . In words, f γ is a composition of projections: first from Z i to Z i , then from Z i to Z i , and finally from Z i K back to Z i .The following theorem gives a sufficient criterion for a complex numberto be approximated by Galois conjugates of stretch factors arising fromPenner’s construction using a curve collection C . Theorem 2.1.
Let C be a collection of curves satisfying the hypotheses ofPenner’s construction and let Ω = i ( C, C ) . Let γ be a closed path in G (Ω) (not necessarily traversing every vertex). If θ is an eigenvalue of f γ and itis not an algebraic unit, then θ ∈ GP ( C ) . The main ingredient of the proof is a result from the paper [Str16]. Beforewe state the theorem, we first recall some notations from Section 2.3 of thatpaper.Associated to the Dehn twists about the curves c i are n × n integralmatrices Q i (depending only on Ω) with the following property: for a productof the Dehn twists about the c i where every twist appears at least once, the ALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCH FACTORS 5 corresponding product M of the Q i is a Perron–Frobenius matrix whoseleading eigenvalue equals the stretch factor of the pseudo-Anosov map.The following is the combination of Lemma 1.2 and Theorem 3.1 of [Str16]. Theorem 2.2.
Let Ω be the intersection matrix of a collection of curvessatisfying the hypotheses of Penner’s construction. Let γ = ( i . . . i K i ) bea closed path in G (Ω) visiting each vertex at least once. Let (2.1) M γ,k = Q ki K · · · Q ki and let λ k be the Perron–Frobenius eigenvalue of M γ,k . Denote by u k ( x ) and v ( x ) the characteristic polynomials χ ( M γ,k ) and χ ( f γ ) , respectively. Thenwe have u k ( x ) x − λ k → v ( x ) . If, in addition, v ( θ ) = 0 and θ k → θ is a sequence such that u k ( θ k ) = 0 and θ k = θ for all but finitely many k , then θ k and λ k are Galois conjugatesfor all but finitely many k . We are now ready to prove Theorem 2.1.
Proof of Theorem 2.1.
Since χ ( f γ ) is invariant under homotopy of γ [Str16,Proposition 4.1] and the graph G (Ω) is connected, we may assume that γ = ( i . . . i K i ) traverses every vertex. Then each matrix M γ,k correspondsto a pseudo-Anosov mapping class with stretch factor λ k .The characteristic polynomials u k ( x ) of M γ,k are monic and have constantcoefficient ±
1, because the matrices Q i are invertible. This can be seendirectly from the definition of the matrices Q i in Section 2.3 of [Str16]. Sothe roots of u k ( x ) are algebraic units.By the first part of Theorem 2.2, there is sequence θ k → θ such that u ( θ k ) = 0 for all k . Since θ is assumed not to be an algebraic unit, we have θ k = θ for all but finitely many k . By the second part of Theorem 2.2,this implies that θ k is a Galois conjugate of λ k . So the number θ is indeedapproximated by Galois conjugates of Penner stretch factors arising fromthe collection C . (cid:3) Approximation of linear maps by compositions of projections
In this section we define rich collections of curves and prove that if C issuch a collection of curves, then we have GP ( C ) = C . This will reduce ourmain theorem to the problem of constructing rich collections of curves onvarious surfaces. First, we need the following definitions.We define the cross-ratio of a 2 × M = (cid:18) m m m m (cid:19) by the formula cr( M ) = m m m m . In order for the cross-ratio to be defined,all matrices are assumed to have positive entries throughout this section. BAL ´AZS STRENNER
Denote by R × + the multiplicative group of the positive reals. The cross-ratio group CRG( M ) of a matrix M is the subgroup of R × + generated by thecross-ratios of all 2 × M . Note that any subgroup of R × + iseither trivial, infinite cyclic or dense. Definition 3.1.
We call a collection of curves C = { c , . . . , c n } on a surface S rich if • C fills S , • n ≥ C = { c , c , c } and C = { c , c , c } form multicurves, • i ( C , C ) has positive entries, rank 3 and dense cross-ratio group. Theorem 3.2 (Criterion for density of Galois conjugates) . If C is a richcollection of curves on S , then GP ( C ) = C . The following subsections develop material necessary for the proof. Theproof will be given at the end of the section.3.1.
Bipartite subgraphs.
Suppose 1 ≤ k < k ≤ n and let I = { , . . . , k } J = { k + 1 , . . . , k } . Suppose that ω ij = 0 whenever i, j ∈ I or i, j ∈ J , and ω ij > i ∈ I, j ∈ J or j ∈ I, i ∈ J . In other words, we assume that the subgraphof G (Ω) spanned by the vertices I ∪ J is a complete bipartite graph. Com-pare the setting I = { , , } and J = { , , } with the definition of richcollections of curves.Define the subspace V = h e k +1 , . . . , e k i generated by the standard basis vectors indexed by J . For 1 ≤ i ≤ k , thesubspace X i = V ∩ Z i is a hyperplane in V , because Z i is hyperplane in R n and V Z i . When i ∈ I and j ∈ J , the projection p i ← j restricts to V and induces a projection s i ← j : V → X i in the direction of e j . On the other hand, the restriction s j ← i of p j ← i on V is the identity.Let γ = ( i j . . . i K j K i ) be a closed path in G (Ω), starting at i ∈ I , andassume that it only traverses the vertices in I ∪ J . Then f γ induces a linearendomorphism s γ : X i → X i by the formula s γ = ( s i ← j K ◦ s j K ← i K ◦ · · · ◦ s i ← j ◦ s j ← i ) | X i , which simplifies to s γ = ( s i ← j K ◦ · · · ◦ s i ← j ◦ s i ← j ) | X i , since the omitted terms are the identity maps. Since s γ is simply the re-striction of f γ to the invariant subspace X i , we have the following. ALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCH FACTORS 7
Proposition 3.3.
The characteristic polynomial of s γ divides the charac-teristic polynomial of f γ . In other words, the eigenvalues of s γ form a subset of the eigenvalues of f γ . Thus having control over the eigenvalues of s γ is useful for applyingTheorem 2.1.In the proof of [Str16, Proposition 4.1], it was shown that f γ is invariantunder homotopies of γ that fix the last edge of γ . This property is inheritedby s γ . In fact a stronger homotopy invariance holds for s γ : it is invariantunder all homotopies fixing the base point i , without the assumption thatthe last edge of γ is fixed throughout the homotopy. To see this we onlyneed to check that the removal of the backtracking i K j K i , when i K = i ,from γ does not change s γ . This is because only the projection s i ← j K | X iK is dropped from the composition, but it is a projection from X i to X i soit does not have any effect.As a consequence, the map γ s γ induces a well-defined map ρ i : π ( G ′ , i ) → End( X i )where G ′ is the subgraph of G (Ω) spanned by the vertex set I ∪ J andEnd( X i ) is the set of linear endomorphisms of X i . Moreover, this map isan anti-homomorphism: s γ ∗ γ = s γ ◦ s γ . This property reduces the computation of s γ for a long path γ to the com-putation of s γ for short paths γ . It also shows that the image of ρ i is infact in GL( X i ).3.2. An example.
Let I = 1 , J = 3 ,
4. The upper left 4 × (cid:18) YY T (cid:19) , where Y = (cid:18) a bc d (cid:19) is a 2 × V is the 2-dimensional subspace generated by e and e . The hyperplanes Z and Z are the orthogonal complements of firstand second rows of Ω. Hence X and X are the lines in V with equations a e + b e = 0 and c e + d e = 0. The slopes are − ba and − dc , respectively.The lines are illustrated on Figure 3.1.Let γ be some non-contractible closed path of length 4 in the subgraph G ′ of G (Ω) spanned by I ∪ J . For instance, let γ = (14231). Then s γ = s ← | X ◦ s ← | X . The projection s ← | X : X → X changes only the e -coordinate of points, and it changes it by a factor adbc which is the ratioof the slopes of X and X . The projection s ← | X then projects back onto X without changing the e -coordinate. Hence the composition is a scalingof the line X by a factor adbc . Therefore adbc is an eigenvalue of s γ and f γ .The graph G ′ is topologically a circle and ρ : π ( G ′ , → End( X ) mapsthe generator of the infinite cyclic group to the scaling of X by adbc . As aconsequence, the eigenvalues of s γ for closed paths γ in G ′ are precisely theinteger powers of adbc . BAL ´AZS STRENNER V e e X X Figure 3.1.
The map s γ : X → X .3.3. Short closed paths.
The content of this section is a geometric de-scription of s γ when γ is a path of length 4. Section 3.2 discussed the simplespecial case when G ′ is a graph on four vertices. In this section we allow G ′ to be bigger, hence V to have dimension greater than 2.Consider a closed path ( i j i j i ) where i , i ∈ I and j , j ∈ J . Asso-ciated to ( i j i j i ) are the following data: • Ω j j i i = (cid:18) ω i j ω i j ω i j ω i j (cid:19) , • c j j i i = cr(Ω j j i i ), and • s j j i i = s ( i j i j i ) = s i ← j | X i ◦ s i ← j | X i ∈ GL( X i ).Note that (cid:0) s j j i i (cid:1) − = s j j i i , since ( i j i j i ) and ( i j i j i ) representinverse elements in π ( G ′ , i ). So the monoid generated by the linear maps s j j i i ∈ GL( X i ), where ( i j i j i ) runs through all closed paths of length4 in G ′ with base point i , is actually a group. Proposition 3.4.
Suppose ( i j i j i ) is a homotopically nontrivial closedpath in G ′ . Then(i) s j j i i acts as the identity on F = X i ∩ X i ;(ii) The 2-dimensional subspace h e j , e j i is not contained in the hyperplane X i , therefore L = X i ∩ h e j , e j i is a line;(iii) s j j i i stretches the line L by a factor of c j j i i .Moreover, if c j j i i = 1 , then F is a codimension 1 subspace in X i , and wehave X i = F ⊕ L .Proof. We first prove statement (i). The map s j j i i is the composition of aprojection from X i to X i and a projection from X i to X i , both of whichact on X i ∩ X i as the identity.For part (ii), note that j = j , otherwise ( i j i j i ) is contractible.Hence h e j , e j i is a 2-dimensional subspace. Since ω i j = 0 and ω i j = 0, ALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCH FACTORS 9 the vectors e j and e j are not orthogonal to the row vector e Ti Ω, hencethey are not contained in X i .For part (iii), observe that the line X i ∩ h e j , e j i is generated by v = ω i j e j − ω i j e j . Since s i ← j is a projection on X i in the direction of e j ,we have s i ← j ( v ) = ω i j ω i j ( ω i j e j − ω i j e j ) . Similarly, s i ← j ( s i ← j ( v )) = ω i j ω i j ω i j ω i j ( ω i j e j − ω i j e j ) = c j j i i v . Finally, the condition c j j i i = 1 implies that the rows of Ω j j i i are notconstant multiples of each other, hence X i = X i . Since both X i and X i are hyperplanes in V , their intersection has codimension 1 in X i . (cid:3) In summary, the linear map s γ takes a very simple form when γ =( i j i j i ) is a closed path with c j j i i = 1: it fixes a hyperplane and stretchesa line by the positive factor c j j i i . In the next section we consider these linearmaps as building blocks for constructing more complicated linear maps.3.4. Linear endomorphisms of planes.
Let W be a 2-dimensional vectorspace over R . For any v ∈ W and a > • Fix( v , a ) ⊂ GL + ( W ) the group of linear maps fixing the vector v and having determinant a n for some integer n ; • Fix( v ) ⊂ GL + ( W ) the group of linear maps fixing the vector v andhaving positive determinant. Lemma 3.5.
Let v ∈ W and f , f ∈ Fix( v ) . Suppose that f and f havelinearly independent eigenvectors w and w with eigenvalues a > and a > , respectively. Then Fix( v , a ) and Fix( v , a ) are contained in theclosure h f , f i ⊂ GL + ( W ) .Proof. In the basis ( w , v ) the maps f and f are described by the matrices A = (cid:18) a
00 1 (cid:19) and A = (cid:18) a b (cid:19) where b = 0. Define C n = A n A A − n = a ba n ! and let C n +1 C − n = b (1 − a ) a n +11 a ! . The bottom left entry of C n +1 C − n tends to zero, therefore powers of C n +1 C − n are dense in the subgroup of matrices of the form (cid:18) ∗ (cid:19) . Finally, note that multiplying these matrices by powers of A i yields ev-erything in Fix( v , a i ) = (cid:26)(cid:18) a ni t (cid:19) : n ∈ Z , t ∈ R (cid:27) for i = 1 , (cid:3) Lemma 3.6.
Let v , v ∈ W be linearly independent vectors, and let a , a > such that h a , a i is dense in R × + . Then h Fix( v , a ) , Fix( v , a ) i = GL + ( W ) . Proof.
Under the action of Fix( v , a ), the orbit of any vector that is nota constant multiple of v is a collection of lines parallel to v . A similarstatement is true for Fix( v , a ). Since v and v are linearly independent,it follows that H = h Fix( v , a ) , Fix( v , a ) i acts transitively on nonzero vectors.Conjugating Fix( v , a ) by an element of H mapping v to v shows thatFix( v , a ) ⊂ H . Since h a , a i is dense in R × + , we have h Fix( v , a ) , Fix( v , a ) i = Fix( v )and Fix( v ) ⊂ H .Finally, notice that any f ∈ GL + ( W ) can be written as f ◦ f where f ∈ H sends v to f ( v ) and f ∈ Fix( v ). Hence H = GL + ( W ). (cid:3) Cross-ratio groups of matrices.
The final ingredient for the proofof Theorem 3.2 is Lemma 3.10 below, which relates dense cross-ratio groupsto the density of eigenvalues of the maps f γ in the complex plane. Lemma 3.7. If M is a × matrix and M i denotes its × submatrixobtained by deleting the i th column, then cr( M )cr( M ) = cr( M ) . Proof. cr( M )cr( M ) = m m m m · m m m m = m m m m = cr( M ) . (cid:3) Corollary 3.8. If M is a × matrix with nontrivial cross-ratio group,then the cross-ratio of at least two of its × submatrices is not 1. Proposition 3.9.
Let M be a × matrix of full rank and dense cross-ratiogroup. Then M has two × submatrices with the following two properties:(i) they are not contained in the same two rows or the same two columns(ii) their cross-ratios generate a dense subgroup of R × + . ALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCH FACTORS 11
Proof.
Since M has dense cross-ratio group, there are two 2 × M and M that satisfy (ii). If they also satisfy (i), then we are done, soassume for example that they are in the same two columns. Then we canreplace M by another 2 × × M would have cross-ratios that are rational powers of cr( M ), and byLemma 3.7 the same would be true for cr( M ). (cid:3) Lemma 3.10.
Suppose that the upper left × submatrix of Ω has the form (cid:18) YY T (cid:19) for a × matrix Y with positive entries such that rank( Y ) = 3 and CRG( Y ) is dense in R × + .Let ε > and u ( x ) = x + ax + b ∈ R [ x ] be arbitrary where b > . Thenthere exists a closed path γ in G (Ω) , and v ( x ) = x + a ′ x + b ′ ∈ R [ x ] with | a − a ′ | < ε and | b − b ′ | < ε such that v ( x ) | χ ( f γ ) .Proof. We assume the notations of Sections 3.1 and 3.3 with I = { , , } and J = { , , } . Note that X is a 2-dimensional subspace in V . ByProposition 3.9, we may assume without loss of generality that c and c are not rational powers of each other.For all i ∈ { , } and j, j ′ ∈ { , , } the map s jj ′ i ∈ GL + ( X ) acts as theidentity on the line F i = X i ∩ X and stretches the line L jj ′ = X ∩ h e j , e j ′ i by c jj ′ i by Proposition 3.4. Note that the fact that Y has full rank impliesthat F and F are distinct and L , L and L are pairwise distinct.Let H be the submonoid of GL + ( X ) generated by the maps s jj ′ i ∈ GL + ( X ) where i ∈ { , } and j, j ′ ∈ { , , } . It is in fact a subgroup,since s jj ′ i and s j ′ j i are inverses of each other (cf. Section 3.3).The maps s , s and s are elements of GL + ( X ) fixing the line F and stretching along the lines L , L and L , respectively. Recall thatthe indices were chosen so that c = 1, so by Corollary 3.8, at least twoof the stretch factors c , c and c are different from 1. By Lemma 3.5,Fix( F , c ) is contained in H . By applying the same reasoning for the maps s , s and s , we get that Fix( F , c ) is also contained in H . Lemma 3.6then implies H = GL + ( X ) . Now pick an element g ∈ GL + ( X ) with χ ( g ) = u ( x ). Then pick an h ∈ H sufficiently close to g so that χ ( h ) = x + a ′ x + b ′ satisfies | a − a ′ | < ε and | b − b ′ | < ε . By the definition of H , there is a closed path γ with basepoint 1 visiting only the vertices 1 , , , , h = s γ . So thestatement follows by Proposition 3.3. (cid:3) Proof of Theorem 3.2.
Let Ω = i ( C, C ) and let θ ∈ Cr { } be arbitrary.Then v ( x ) = ( x − θ )( x − ¯ θ ) ∈ R [ x ] has positive constant term, so byLemma 3.10 there exist • a sequence ( v j ( x )) j ∈ N of monic quadratic polynomials in R [ x ] and • a sequence ( γ j ) j ∈ N of closed paths in G (Ω)such that • v j ( x ) → v ( x ) and • v j ( x ) | χ ( f γ j ) for every j ∈ N .Let ( θ j ) j ∈ N be a sequence such that v j ( θ j ) = 0 for all j ∈ N and θ j → θ .Assume for a moment that θ is not an algebraic unit of degree at most2, that is, θ is not a root of a polynomial x + sx ± s ∈ Z . Inthis case, θ j is not an algebraic unit if j is large enough, because the setof algebraic units of degree at most 2 is a discrete subset of C . Hence byTheorem 2.1, we have θ ∈ GP ( C ).To complete the proof, note that the set of θ ∈ C where θ = 0 and θ is not an algebraic unit of degree at most 2 is dense in C so we have GP ( C ) = C . (cid:3) Construction of rich collection of curves
In this section, we show that rich collections of curves exist on sufficientlycomplicated surfaces.
Proposition 4.1. If ξ ( S ) ≥ , then there exists a rich collection of curveson S . First we prove a lemma about intersection matrices. For a multicurve B = { b , . . . , b ℓ } on S and a vector s = ( s , . . . , s ℓ ) with integer coordinates,the product T s B = Q ℓj =1 T s j b j is called a multitwist about the multicurve B . Lemma 4.2.
Let A and B be multicurves of S . If s > or s < , then i ( A, T s B ( A )) = i ( A, B ) D | s | i ( B, A ) where D | s | is the ℓ × ℓ diagonal matrix with entries | s | , . . . , | s ℓ | on the diag-onal.Proof. If A, B and C are multicurves on S and s > s <
0, then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ( a, T s B ( c )) − ℓ X j =1 | s j | i ( a, b j ) i ( b j , c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ i ( a, c )for all a ∈ A and c ∈ C [FM12, Prop. 3.4]. We can summarize theseinequalities in the single inequality(4.1) (cid:12)(cid:12) i ( A, T s B ( C )) − i ( A, B ) D | s | i ( B, C ) (cid:12)(cid:12) ≤ i ( A, C ) . We obtain the claimed equation by setting C = A . (cid:3) Proof of Proposition 4.1.
Consider the pairs of multicurves A , B on S , , S , and S , , pictured on Figure 4.1. In the three cases, the intersection ALOIS CONJUGATES OF PSEUDO-ANOSOV STRETCH FACTORS 13
Figure 4.1.
Curvesmatrix i ( A, B ) is , and , respectively. In the first two cases i ( A, B ) i ( B, A ) is
12 8 48 8 44 4 4 and . In the third case i ( A, B ) i ( B, A ) = . By Lemma 4.2, we obtain a pair of multicurves on all three surfaces withrank 3 intersection matrix with positive entries and dense cross-ratio group.The curves necessarily fill the surface in each case, so we obtain a richcollection of curves.Any other compact orientable surface with ξ ( S ) ≥ C on all these surfaces thatsatisfy all properties of richness except the filling property. However, thefilling property is easily achieved by extending both multicurves to maximalmulticurves, being careful not to include the same curve in both maximalmulticurves. (cid:3) We are now ready to prove Theorem 1.2.
Theorem 1.2. If S is a compact orientable surface with ξ ( S ) ≥ , thenthere is a collection of curves C on S such that GP ( C ) = C .Proof. There exists a rich collection of curves by Proposition 4.1. By The-orem 3.2, this implies that GP ( C ) = C . (cid:3) Acknowledgements.
We are grateful to Ursula Hamenst¨adt, Autumn Kent,Dan Margalit and the referees for their comments and help.
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