Typical and atypical properties of periodic Teichmüller geodesics
Consider a component of a stratum in the moduli space of area one abelian differentials on a surface of genus . Call a property for periodic orbits of the Teichmüller flow on typical if the growth rate of orbits with property is maximal. Typical are: The logarithms of the eigenvalues of the symplectic matrix defined by the orbit are arbitrarily close to the Lyapunov exponents of , and its trace field is a totally real splitting field of degree over . If then periodic orbits whose -orbit closure equals are typical. We also show that contains only finitely many algebraically primitive Teichmüller curves, and only finitely many affine invariant submanifolds of rank .
AMS subject classification: 37C40, 37C27, 30F60
Research supported by ERC grant “Moduli”
The mapping class group of a closed surface of genus acts by precomposition of marking on the Teichmüller space of marked complex structures on . The action is properly discontinuous, with quotient the moduli space of complex structures on .
The goal of this paper is to describe properties of this action which are invariant under conjugation and hold true for conjugacy classes of mapping classes which are typical in the following sense.
The moduli space of area one abelian differentials on decomposes into strata of differentials with zeros of given multiplicities. There is a natural -action on any connected component of this moduli space. The action of the diagonal subgroup is the Teichmüller flow .
Let be the set of all periodic orbits for on . The length of a periodic orbit is denoted by . Let be the entropy of the unique -invariant Borel probability measure on in the Lebesgue measure class [M82, V86]. As an application of [EMR12] (see also [EM11, H11]) we showed in [H13] that
Call a subset of typical if
Thus a subset of is typical if its growth rate is maximal. The intersection of two typical subsets is typical.
A periodic orbit for determines the conjugacy class of a pseudo-Anosov mapping class. Each mapping class acts on . This defines a natural surjective [FM12] homomorphism
Thus a periodic orbit of determines the conjugacy class of a matrix .
be the positive Lyapunov exponents of the Kontsevich Zorich cocycle with respect to the normalized Lebesgue measure on . The fact that there are no multiplicities in the Lyapunov spectrum was shown in [AV07]. For let be the logarithm of the absolute value of the -th eigenvalue of the matrix ordered in decreasing order and write . As is symplectic, with real leading eigenvalue , we have
As eigenvalues of matrices are invariant under conjugation, for we obtain in this way a function .
The characteristic polynomial of a symplectic matrix is a reciprocal polynomial of degree with integral coefficients. Its roots define a number field of degree at most over which is a quadratic extension of the so-called trace field of . The Galois group of is isomorphic to a subgroup of the semi-direct product where is the symmetric group in variables (see [VV02] for details). The field and the Galois group only depend on the conjugacy class of .
For let be the Galois group of the number field defined by the conjugacy class . We show
For the set is typical.
The set of all such that the trace field of is totally real, of degree over , and is typical.
The proof of Theorem 1 uses a result on the Zariski closure of the image under the map of pseudo-Anosov mapping classes obtained from the first return map of the Teichmüller flow on to a small contractible flow box in . Although our viewpoint is a bit different, this discussion can be translated into properties of the Rauzy-Veech group of and yields the following result which was conjectured by Zorich [Z99].
The Rauzy-Veech group of any component of a stratum is a Zariski dense subgroup of .
For hyperelliptic strata, Avila, Matheus and Yoccoz [AMY16] showed that the Rauzy-Veech group is a subgroup of of finite index. In Proposition 3.4 we observe that for strata with at least one simple zero, the Rauzy Veech group coincides with . Theorem 6.3 is a more precise version of Corollary 1 which is valid for affine invariant manifolds as well.
By the groundbreaking work of Eskin, Mirzakhani and Mohammadi [EMM15], such affine invariant manifolds are precisely the closures of orbits for the -action on . Examples of non-trivial orbit closures are arithmetic Teichmüller curves. They arise from branched covers of the torus, and they are dense in any stratum of abelian differentials. Other examples of orbit closures different from entire components of strata can be constructed using more general branched coverings.
The rank of an affine invariant manifold is defined by
where is the projection of period coordinates into absolute cohomology [W15]. Teichmüller curves are affine invariant manifolds of rank one, and the rank of a component of a stratum equals .
We establish a finiteness result for affine invariant submanifolds of rank at least two which is independently due to Eskin, Filip and Wright [EFW17].
Let and let be a component of a stratum in the moduli space of abelian differentials. For every , there are only finitely many proper affine invariant submanifolds in of rank .
As a corollary, we obtain
Let be any component of a stratum in genus . Then the set of all whose -orbit closure equals is typical.
For , Corollary 2 is false in a very strong sense. Namely, McMullen [McM03a] showed that in this case, the orbit closure of any periodic orbit is an affine invariant manifold of rank one. If the trace field of the orbit is quadratic, then defines a Hilbert modular surface in the moduli space of principally polarized abelian varieties which contains the image of the orbit closure under the Torelli map. Such a Hilbert modular surface is a quotient of by the lattice where is the ring of algebraic integers in . This insight is the starting point of a complete classification of orbit closures in genus [McM03b].
In higher genus, Apisa [Ap15] classified all orbit closures in hyperelliptic components of strata. For other components of strata, a classification of orbit closures is not available. However, there is substantial recent progress towards a geometric understanding of orbit closures. In particular, Mirzakhani and Wright [MW16] showed that all affine invariant manifolds of maximal rank either are components of strata or are contained in the hyperelliptic locus. We refer to the work [LNW15] of Lanneau, Nguyen and Wright for an excellent recent overview of what is known and for a structural result for rank one affine invariant manifolds.
Teichmüller curves are affine invariant manifolds of dimension . To each such Teichmüller curve, there is associated a trace field which is an algebraic number field of degree at most over . This trace field coincides with the trace field of every periodic orbit contained in the curve [KS00]. The Teichmüller curve is called algebraically primitive if the degree of its trace field equals .
The stratum of abelian differentials with a single zero on a surface of genus 2 contains infinitely many algebraically primitive Teichmüller curves [McM03b]. Recently, Bainbridge, Habegger and Möller [BHM14] showed finiteness of algebraically primitive Teichmüller curves in any stratum in genus . Finiteness of algebraically primitive Teichmüller curves in strata of differentials with a single zero for surfaces of prime genus was established in [MW15]. Our final result generalizes this to every stratum in every genus , with a different proof. A stronger finiteness result is contained in [EFW17].
Any component of a stratum in genus contains only finitely many algebraically primitive Teichmüller curves.
Section 3 introduces the idea of local monodromy groups and their Zarisky closures and uses it to show Corollary 1. In Section 4, this result together with group sieving and the first part of Theorem 1 leads to the second part of Theorem 1 and to Corollary 1.
Section 5 contains some properties of the absolute period foliation of an affine invariant manifold. In Section 6 we look at the local monodromy group of an affine invariant manifold and show that it is Zariski dense in the symplectic group of rank corresponding to the rank of the manifold. We then compare in Section 7 the Chern connection on the Hodge bundle to the Gauss Manin connection which leads to the proof of the first part of Theorem 2 in Secction 8. Section 9 uses information on the absolute period foliation to complete the proof of Theorem 2. The proofs of Theorem 2 and Theorem 3 are contained in Section 8.
Acknowledgement: During the various stages of this work, I obtained generous help from many collegues. I am particularly grateful to Alex Wright for pointing out a mistake in an earlier version of this work. Both Matt Bainbridge and Alex Eskin notified me about parts in an earlier version of the paper which needed clarification. Yves Benoist provided the proof of Proposition 6.8, and discussions with Curtis McMullen inspired me to the differential geometric approach in Section 9. Part of this article is based on work which was supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the MSRI in Berkeley, California, in spring 2015.
2. Lyapunov exponents
In this section we consider any component of a stratum of area one abelian differentials. The results in this section are equally valid for any affine invariant submanifold for the -action [EMM15], in particular they hold true for all components of strata of quadratic differentials.
The Teichmüller flow acts on preserving a Borel probability measure in the Lebesgue measure class, the so-called Masur Veech measure. Let be the entropy of with respect to the measure .
Denote by the trivial vector bundle over with fibre . The Kontsevich Zorich cocycle over is an extension of the Teichmüller flow on to a flow on which can roughly be described as follows. Consider a component of the preimage of in the Teichmüller space of marked area one abelian differentials. Then is the quotient of under the action of the stabilizer of in the mapping class group . For and large , the differential can be brought back to a fixed fundamental domain for the action of by an element . The action of on the first cohomology group is essentially the Kontsevich Zorich cocycle.
The Kontsevich Zorich cocycle can also be described using the Gauss Manin connection on . The Gauss Manin connection is a flat connection on defined by local trivializations which identify nearby integral cohomology classes. Parallel transport for the Gauss Manin connection defines a lift of the Teichmüller flow to a flow on which preserves the symplectic structure on defined by the algebraic intersection form on .
There are some technical difficulties due to nontrivial point stabilizers for the action of the mapping class group on Teichmüller space. To avoid dealing with this issue (although this can be done with some amount of care) we define the good subset of to be the set of all points with the following property. Let be a component of the preimage of in the Teichmüller space of marked abelian differentials and let be a lift of ; then an element of which fixes acts as the identity on (compare [H13] for more information on this technical condition). The good subset is open, dense and -invariant [H13].
The Kontsevich Zorich cocycle is bounded, with values in the symplectic group , and therefore its Lyapunov exponents for the invariant measure are defined. These exponents measure the asymptotic growth rate of vectors along orbits of which are typical for . Since the Gauss Manin connection preserves the symplectic structure on , these exponents are invariant under multiplication with . Let be the largest Lyapunov exponents of the Teichmüller flow on . That these exponents are all positive and pairwise distinct was shown in [AV07]. For more general affine invariant manifolds, the analogue statement need not hold true. We refer to [Au15] for a discussion and examples.
be the countable collection of all periodic orbits for contained in . Denote by the period of . The orbit determines a conjugacy class in of pseudo-Anosov elements. Let be an element in this conjugacy class; then is determined by up to conjugation. Furthermore, the largest absolute value of an eigenvalue of equals . More precisely, the matrix is Perron Frobenius, with leading eigenvalue , and the eigenspace for the eigenvalue is spanned by the real cohomology class defined by intersection with the attracting measured geodesic lamination of .
If we define to be the quotients by of the logarithms of the largest absolute values of the eigenvalues of the matrix , ordered in decreasing order and counted with multiplicities, then the numbers only depend on but not on any choices made.
Let . For define if for every and define otherwise.
For let be the set of all periodic orbits of prime period contained in the interval . For an open or closed subset of denote by the characteristic function of and define
Clearly we have
for all .
Call a point birecurrent if it is contained in its own - and limit set. By the Poincaré recurrence theorem, the set of birecurrent points in has full Lebesgue measure. In [H13] (Corollary 4.8 and Proposition 5.4) we showed
For every good birecurrent point , for every neighborhood of in and for every there is an open neighborhood of in and a number such that
for all sufficiently large .
The proof of Proposition 2.1 is based on a more technical result which will be used several times in the sequel. Lemma 2.2 below combines Lemma 4.7 and Proposition 5.4 of [H13]. For its formulation, we say that a closed curve in defines the conjugacy class of a pseudo-Anosov mapping class if the following holds true. Let be a lift of to an arc in the Teichmüller space of abelian differentials, parametrized one some interval ; then for a mapping class which is conjugate to . This definition does not depend on any choices made.
Let be a good birecurrent point and let . Then every neighborhood of contains an open and contractible neighborhood of , such that there exists a nested sequence of neighborhoods of , with closed and contained in the interior of , and there is a number with the following properties.
Let and let be such that . Let be the connected component containing of the intersection .
The length of the connected subsegment of the orbit containing equals .
, and the Lebesgue measure of the intersection is contained in the interval
Connect to by an arc in and let be the concatenation of the orbit segment with this arc. We call a characteristic curve of the orbit segment . There is a unique periodic orbit for of length at most which intersects . The curve and the orbit define the same conjugacy class in .
Note that in the above statement, we slightly adjusted the choice of the sets compared to the terminology in [H13] for clarity of exposition.
We use Lemma 2.2 to show
For every birecurrent point , for every neighborhood of in and for every there is an open neighborhood of in and a number with the properties stated in Proposition 2.1 such that for every we have
Let be the Hodge norm on the bundle (see [ABEM12] for definitions and the most important properties). Denote as before by the lift of the Teichmüller flow to a flow on defined by parallel transport for the Gauss Manin connection. Recall that preserves the symplectic structure on , but in general it does not preserve the Hodge norm. For let be the fibre of at . For and for let
be the minimum of the operator norms of the restriction of to a symplectic subspace of of real dimension . Define
Let and let be a neighborhood of a birecurrent point . Since the Kontsevich Zorich cocycle is locally constant (or, equivalently, the Gauss Manin connection is flat), we can find a collection of nested neighborhoods with the properties in Lemma 2.2 and such that furthermore, with the notations from the lemma, if and if then the periodic orbit for determined by the characteristic curve of the orbit segment satisfies
Namely, for a sufficiently small contractible neighborhood of in , the trivialization of defined by the Gauss Manin connection almost preserves the Hodge norm. Then the estimate (1) holds true if we replace be the -th absolute value in decreasing order of an eigenvalue of the symplectic transformation of which is defined by parallel transport for the Gauss Manin connection along a characteristic curve for the orbit segment . But by property (c) in Lemma 2.2, the characteristic curve defines the same conjugacy class of a pseudo-Anosov mapping class as . This means that the numbers are precisely the absolute values of the eigenvalues of the transformation . Thus the estimate for the characteristic curve and the transformation implies the estimate for .
By Oseledec’s theorem and ergodicity, there is number and a Borel subset of of measure with the following property. Let and let ; then .
Since the Lebesgue measure is mixing for the Teichmüller flow, there is a number such that
for all . By the estimate (b) in Lemma 2.2, this implies that the number of components of the intersection containing points in is at least . By property (c) in Lemma 2.2, for each such component there is a periodic orbit of passing through . The estimate (1) together with the definition of yields that each such orbit satisfies . Thus by (a) of Lemma 2.2, each such component of intersection contributes to the value . Together this shows the proposition. ∎
As a corollary, we obtain the first part of Theorem 1. We formulate it more generally for strata of abelian or quadratic differentials. As before, denotes the -th Lyapunov exponent of the Kontsevich Zorich cocycle.
For , the set is typical.
In [H13] the following is shown. As , the measures
converge weakly to the Lebesgue measure on . The Lebesgue measure of vanishes, so it suffices to study the Lebesgue measure on .
In fact, there is a compact subset of such that the growth rate of all periodic orbits which do not intersect is strictly smaller than .
3. Local Zariski density: The Zorich conjecture
In this section we prove the Zorich conjecture which is stated as Corollary 1 in the introduction. Throughout the section, we denote by a component of a stratum in the moduli space of area one abelian differentials on , and by a component of its preimage in the Teichmüller space of abelian differentials. As in Section 2, we denote by the open dense -invariant subset of good points. All results of this section are also true for components of quadratic differentials, however the proof requires some more details which we postpone to forthcoming work.
A periodic orbit of on determines a conjugacy class in . However, it will be convenient to look at actual elements of rather than at conjugacy classes.
Choose a birecurrent point . Let and let be a nested family of neighborhoods of in as in Lemma 2.2.
For sufficiently large let and let be such that . A characteristic curve of this orbit segment determines uniquely a periodic orbit of which intersects in an arc of length . There may be more than one such intersection arc, but there is a unique arc determined by the component of the intersection containing the point as described in (c) of Lemma 2.2. Choose the midpoint of this intersection arc as a basepoint for and as an initial point for a parametrization of .
Let be the set of all parametrized periodic orbits of this form for points with . By Lemma 2.2, the map which associates to a component of containing points in the corresponding parametrized periodic orbit in is a bijection.
Fix once and for all a lift of the contractible set to . A periodic orbit which intersects in an arc of length lifts to a subarc of a flow line of the Teichmüller flow on with starting point in . The endpoints of this arc are identified by a pseudo-Anosov element .
The following shadowing property is a variation of Lemma 2.2, using the same notations. Versions of this lemma are familiar in hyperbolic dynamics.
For , there is a point , and there are numbers with the following properties.
For each , a characteristic curve of the orbit segment defines the same conjugacy class in as the periodic orbit .
There is a parametrized periodic orbit for with initial point in which defines the same conjugacy class in as a characteristic curve of the orbit segment .
In the case that the arcs are contained in a fixed compact invariant subset for and that the set is chosen small in dependence of , the lemma is identical with the slight weakening of Theorem 4.3 of [H10]. That the statement holds true in the form presented here is immediate from the construction of the set in [H13] and Proposition 5.4 of [H13] which is based on an extension of Theorem 4.3 of [H10] to arbitrary orbits of the Teichmüller flow which recur to . ∎
As a consequence, the subsemigroup of generated by consists of pseudo-Anosov elements whose corresponding periodic orbits are contained in the stratum and pass through the set . This can be viewed as a version of Rauzy-Veech induction as used in [AV07, AMY16] which is valid for strata of quadratic differentials as well, or as a version of symbolic dynamics for the Teichmüller flow on strata as in [H11, H16].
Our next goal is to obtain information on the image of this subsemigroup under the homomorphism . For this we choose an odd prime and study the image of under the natural reduction map
where is the field with elements. Denote by the symplectic form on which is preserved by .
The following lemma relies on the results in [Hl08]. For its formulation, define a transvection in to be a map which fixes a subspace of of codimension one and has determinant one (see [Hl08]). Any map of the form
for some (here as before, is the symplectic form) is a transvection. We call this map a transvection by .
Let be an odd prime and let be a subgroup generated by transvections by the elements of a set which spans . Assume that there is no nontrivial partition so that for all . Then .
For each write . Let be the subgroup generated by the transvections . Since the vectors span , the intersection of the invariant subspaces of the transvections is trivial.
We claim that the standard representation of on is irreducible. Namely, assume to the contrary that there is an invariant proper linear subspace . Let ; then there is at least one so that . By invariance, we have and hence since is a field.
As a consequence, is spanned by some of the , say by , and if is such that for some then . However, this implies that by the assumption on the set .
To summarize, is an irreducible subgroup of generated by transvections (where irreducible means that the standard representation of on is irreducible). Furthermore, as is an odd prime by assumption, the order of each of these transvections is not divisible by . Theorem 3.1 of [Hl08] now yields that which is what we wanted to show. ∎
For an odd prime the subgroup of generated by equals the entire group .
If is a stratum of abelian differentials with at least one simple zero then the subgroup of generated by equals the entire mapping class group.
We begin with showing the second part of the proposition.
The mapping class group is generated by finitely many Dehn twists about simple closed curves. A specific generating system are the so-called Humphries generators (see p.112 of [FM12]). These generators are Dehn twists about the set of simple closed curves shown in the Figure 1.
Let be a stratum of abelian differentials with at least one simple zero. By [KZ03], is connected. For a simple closed curve denote by the positive Dehn twist about .
For the second part of the proposition it suffices to show that for each of the simple closed curves shown in Figure 1, there is so that the subgroup of generated by contains .
By [H16], to each component of a stratum there is associated a collection of large train tracks which parametrize the component in a sense described in [H16]. In particular, if and if is a pseudo-Anosov mapping class with train track expansion (this means that the train track is carried by , a property which we denote by in the sequel), then the periodic orbit of the Teichmüller flow corresponding to the conjugacy class of is contained in the closure of . This statement can be viewed as a version of Rauzy Veech induction.
If is a component of a stratum and if is contained in the closure of , then any train track associated to can be obtained from some train track associated to by removing some branches [H16]. Furthermore, if we call a train track orientable if there exists a consistent orientation of the branches of (here consistent means that the orientation is compatible at the switches), then orientable train tracks correspond to strata of abelian differentials.
Let now be the stratum of abelian differentials with one simple zero and one zero of order . We claim that there is a train track for with the following property. For each of the Humphries generators of there exists such that we have , ie the train track is carried by .
Let as before be the intersection form on . Denote by the homology class of an oriented simple closed curve . Orient the curves in Figure 1 in such a way that for each we have and . For example, we can find such an orientation so that the curves in Figure 1 are oriented counter-clockwise, and the curves clockwise.
Construct from this oriented curve system a train track by replacing each intersection of oriented curves by a large branch as shown in Figure 2.
Informally, this amounts to following the oriented curves in the direction prescribed by their orientation. It is easy to check that is orientable.
Each of the curves is embedded in , and the image of under either the positive or the negative Dehn twist about any of these curves is carried by . Namely, the curves are embedded as a subgraph which either consists of a single large branch and a single small branch (in the case of the curves ) or of two large branches and two small branches as shown in Figure 3 below, or of three large branches and three small branches (for the curves ). For a simple closed curve as shown in Figure 3, the train track obtained from by a single positive Dehn twist about is obtained from two splits at the two large branches shown in the figure.
The train track has two complementary components, one of them is a four-gon. It is not hard to see that is large in the sense of [H16] (ie it is birecurrent, and it carries a large geodesic lamination of the same combinatorial type as ). Thus is associated to the stratum of differentials with one simple zero and one zero of order [H16]. By construction, it has the properties required in the above claim.
A stratum of abelian differentials with a simple zero contains in its closure [KZ03]. There is a train track for so that can be obtained from by removing some branches [H16], i.e. is a subtrack of . We can choose in such a way that for each of the Humphries generators , we have . Thus is a train track as required in the above claim (see again [H16] for details of this construction).
Choose a pseudo-Anosov mapping class which admits as a train track expansion. This means that . Assume that maps every branch of onto . This will guarantee that the periodic orbit defined by is contained in rather than in the boundary of , see [H16]. Then for each of the Humphries generators , the composition satisfies , moreover maps every branch of onto . But this just means that is a pseudo-Anosov mapping class which admits as a train track expansion. In particular, determines a periodic orbit in . We may assume without loss of generality that this orbit is contained in .
We next show that for every periodic orbit in defined by the conjugacy class of a pseudo-Anosov mapping class with train track expansion as above and for every neighborhood of (see the proof of Proposition 2.3), the subgroup of generated by the parametrized orbits in the set contains each of the Humphries generators and hence this group is the entire mapping class group.
Namely, write for one of these Dehn twists. By the above discussion, for each the mapping class is pseudo-Anosov, with train track expansion , and it defines a periodic orbit in . As , the normalized -invariant measures supported on the periodic orbits converge weakly to the normalized Lebesgue measure on . Namely, the vertical projective measured laminations of abelian differentials which generated the periodic orbit are all carried by , and in the space of projective transverse measures on . By the same reasoning, the horizontal projective measured geodesic laminations of converge to the projective measured geodesic lamination of .
Thus for sufficiently large the periodic orbit passes through the set used for the construction of the set and hence it defines an element of . Then is contained in the group generated by as claimed. As a consequence, the second part of the proposition holds true for any choice of a neighborhood of .
To show the second part of the proposition for an arbitrary open neighborhood in of a birecurrent point , recall that by ergodicity of the Teichmüller flow and the Anosov closing lemma established in [H13], “generic” periodic orbits for passing through become equidistributed for the Lebesgue measure (see [H13] for a detailed discussion). In particular, they pass through the set chosen as above.
Now use the argument for the set as follows. Let be the set of periodic orbits constructed from and the open neighborhood of . Let be a generic periodic orbit which passes through the set . Let be such that . Let be the reparametrization of which satisfies . Apply the above construction to and the pseudo-Anosov mapping class defined by the parametrized orbit . We conclude that for a sufficiently large , a reparametrization of the periodic orbit corresponding to is contained in . By the reasoning used for the set , we obtain the second part of the proposition for periodic orbits passing through .
The first part of this proof also immediately implies the first part of the proposition for strata of abelian differentials with at least one simple zero. We are left with showing the first part of the proposition for arbitrary strata of abelian differentials.
Let be an oriented non-separating simple closed curve on which defines the homology class . The action on homology of the positive Dehn twist about equals
(Proposition 6.3 of [FM12]). In other words, the Dehn twist acts on as a transvection by .
By the main result of [KZ03], for the stratum of abelian differentials with a single zero consists of three connected components. One of these components is hyperelliptic, the other two components are distinguished by the parity of the spin structure they define. The stratum consists of two components; one component is hyperelliptic, the second component has odd spin structure. The stratum is connected.
Any component of a stratum with more than one zero contains a component of in its closure. Thus following the reasoning in the first part of this proof, it suffices to find for each of the components of a train track associated to with the following property. Let be an odd prime. Then the subgroup of which is generated by those transvections in which are images under of Dehn twists about embedded curves in with is all of .
Let again be the train track constructed in the beginning of this proof from the Humphries generators of . Let be the train track obtained from by removing the small branch contained in the curve . This train track is orientable, filling and birecurrent. If then is not invariant under a hyperelliptic involution. Hence it corresponds to one of the two non-hyperelliptic components of . These components are distinguished by the parity of the spin structure they define. The formula in the proof of Proposition 4.9 of [H16] calculates this parity (see also the formulas in [KZ03]). It is odd if is odd, and even if is even.
For each curve we have . Now for any choice of orientations of the curves in , the homology classes are a basis of . Moreover, any two curves intersect in at most one point and their union is a connected subset of . This implies that the transvections by the homology classes of these curves satisfy the assumptions in Lemma 3.2. As a consequence, for each odd prime these transvections generate . This is what we wanted to show.
The other two components of are treated in the same way. A train track for the hyperelliptic component of can be constructed from the following curve system. Remove the curves from the Humphries generators and add a simple closed curve which intersects in a single point and does not intersect any other of the curves shown in Figure 1. The orientation of the curve can be chosen in such a way that all the properties used above hold true. The resulting set of curves is clearly invariant under the hyperelliptic involution. The same argument as for the non-hyperelliptic components discussed above yields the case of the hyperelliptic component. In particular, we established the proposition for [KZ03].
We are left with finding for a train track for the second non-hyperelliptic component of with parity of the spin structure opposite to the parity of . Thus let be the component of abelian differentials in genus with a single zero with parity of the spin structure mod 2. By Lemma 14 of [KZ03], there are differentials which can be obtained from an abelian differential with a single zero in genus by “bubbling a handle”. This can be translated into train tracks as follows. Start with a train track on a surface of genus constructed above for the non-hyperelliptic component with a single zero in genus whose spin structure equals mod . Attach to a cylinder by removing from two small disks. There is no ambiguity here since is connected. Following the strategy in [H16] we extend the train track by adding first an embedded simple closed curve defining the core curve of . Attach two small branches to , one at each side of , such that after adding these branches, consists of a single large branch and a single small branch. Connect the branch to the curve and connect the branch to the curve (notations are as in Figure 1) using the above orientation rule. It is now easy to check that the resulting train track is large and defines the stratum of differentials with parity of spin structure mod 2. Furthermore, carries a system of embedded simple closed curves defining a basis for with the properties stated in Lemma 3.2. For each there is a train track carried by and a choice of a sign such that . The reasoning in the beginning of this proof can now be applied to suitably chosen pseudo-Anosov mapping classes with train track expansion so that . This completes the proof of the proposition. ∎
The first part of the following corollary establishes Zorich’s conjecture [Z99] (we leave the easy translation into the language of Rauzy induction to the reader). For its formulation, call a component of a stratum locally Zarisky dense if the following holds true. Let be any open subset of and let be the sub-semigroup of generated in the sense discussed above by the periodic orbits for passing through . We require that the sub-semigroup of is Zariski dense in .
A component of a stratum is locally Zariski dense.
If is a stratum of abelian differentials with a simple zero then the preimage of in the Teichmüller space of area one abelian differentials is connected.
The first part of the corollary follows from the first part of Proposition 3.4 and the well known fact that a subgroup of which surjects onto for all odd primes is Zariski dense in (see [Lu99] for more and for references).
Now let be a stratum of abelian or quadratic differentials with a simple zero and let be a component of the preimage of in the Teichmüller space of abelian differentials. By the second part of Proposition 3.4, the stabilizer of in the mapping class group equals the entire mapping class group. As the components of the preimage of are permuted by the mapping class group, the second part of the corollary follows. ∎
More generally, one can ask about the orbifold fundamental group of a component of a stratum of abelian or quadratic differentials. If is a stratum with zeros then the zero forgetful map maps this orbifold fundamental group into the mapping class group of . Proposition 3.4 shows that for strata with a simple zero, this map is onto. For some strata in genus 3, the fundamental group has been identified with tools from algebraic geometry in [LM14].
For most components of strata, we do not even know the image of the orbifold fundamental group in besides surjecting onto for all odd primes , see Proposition 3.4. For hyperelliptic components, this image has been determined in [AMY16]. One finds that the group is precisely the subgroup stabilized by the hyperelliptic involution, in particular it is of finite index in . We conjecture that the latter property holds true for all components of all strata.
4. Galois groups
In this section we consider again an arbitrary component of a stratum of abelian differentials. We continue to use the assumptions and notations from section 2 and Section 3. Recall in particular the construction of the set of parametrized periodic orbits in defined by a small neighborhood of a point which is birecurrent under the Teichmüller flow. We showed in Section 3 that this set determines a sub-semigroup of consisting of pseudo-Anosov elements whose image under the homomorphism is Zariski dense . More precisely, by the first part of Proposition 3.4, for every odd prime the image of the semi-group generates the entire group . Here as before, denotes reduction modulo .
Now is a finite group and therefore for every there is some such that . As a consequence, for all we have as well and hence is a group. Then equals the group generated by and thus .
Our next goal is to makes this statement quantitative. To this end denote for a periodic orbit for by the -invariant measure supported on whose total mass equals the period of . Recall that periodic orbits in the set are parametrized, so a single unparametrized periodic orbit may give rise to many different elements of .
Let as before be an odd prime and let be the number of elements of . The following proposition holds true for any finite group of order with the property that there is a homomorphism whose restriction to the semigroup is surjective.
Let be arbitrary, let be an odd prime and define
Then as ,
independent of up to a multiplicative error which only depends on and which can be arranged to be arbitrarily close to one.
We show first that there is a number such that
for all and for all sufficiently large .
To this end let be any neighborhood of a birecurrent point . Let and let be as in Lemma 2.2. Let be the sub-semigroup of generated by . By Lemma 3.1, this semigroup consists of pseudo-Anosov elements. Furthermore, each is represented by a parametrized periodic orbit for which intersects the set in a segment of length containing as its midpoint. Vice versa, every periodic orbit which passes through admits a parametrization so that the corresponding element of is contained in . The sub-semigroup of is mapped by onto the finite group .
Since is a finite group and the above argument applies to every neighborhood of in , in particular to , there is a number with the following property. Let be arbitrary. Then there is some with for some