Normal generators for mapping class groups are abundant

Normal generators for mapping class groups are abundant

Abstract.

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.

1. Introduction

Let denote a connected, closed, orientable surface of genus . The mapping class group is the group of homotopy classes of orientation-preserving homeomorphisms of . The goal of this paper is to give new examples of elements of that have normal closure equal to the whole group; in this case we say that the element normally generates .

In the 1960s Lickorish [27] and Mumford [34] proved that is normally generated by a Dehn twist about a nonseparating curve in . On the other hand for the th power of a Dehn twist is not a normal generator since it acts trivially on the mod homology of .

The Nielsen–Thurston classification theorem for mapping class groups categorizes elements of as either periodic, reducible, or pseudo-Anosov; see [12, Chapter 13]. Dehn twists and their powers are examples of reducible elements. Our primary focus in this paper is to find normal generators for among the periodic and pseudo-Anosov elements.

Periodic elements

By the work of Harvey, Korkmaz, McCarthy, Papadopoulos, and Yoshihara [15, 21, 32, 37], there are several specific examples of periodic mapping classes that normally generate . The first author recently showed [23] that for and , there is a mapping class of order that normally generates .

Our first theorem completely answers the question of which periodic elements normally generate. In the statement, the hyperelliptic involution is the element (or, conjugacy class) of depicted in Figure 1.

Theorem 1.

For , every nontrivial periodic mapping class that is not a hyperelliptic involution normally generates .

Additionally, we show in Proposition 3 that for the normal closure of the hyperelliptic involution is the preimage of under the standard symplectic representation of . The Torelli group is the kernel of this representation, so the normal closure of the hyperelliptic involution contains as a subgroup of index 2. We also give a complete classification of the normal closures of periodic elements of when .

Several well-known facts about mapping class groups follow from Theorem 1. For instance, the Torelli group is torsion free. Also, the level congruence subgroup , the kernel of the action of on , is torsion free for . Additionally, it follows that there are no finite nontrivial normal subgroups of except for the cyclic subgroups of and generated by the hyperelliptic involution. One new consequence is that any normal subgroup of not containing is torsion free.

Another consequence of Theorem 1 involves the subgroup of generated by th powers of all elements. Let be the least common multiple of the orders of the periodic elements of . For not divisible by we show that the th power subgroup of is equal to the whole group ; see Corollary 5. This improves on a result of Funar [14], who proved the analogous result with replaced by . Funar’s theorem answers in the negative a question of Ivanov, who asked in his problem list [18, Problem 13] if the th power subgroup of has infinite index for sufficiently large.

Figure 1. Rotation by about the indicated axis is a hyperelliptic involution

Pseudo-Anosov elements with small stretch factor

Having addressed the periodic elements, we turn to the case of pseudo-Anosov mapping classes. In a 1986 paper [28], Darren Long asked:

Can the normal closure of a pseudo-Anosov map ever be all of ?

Long answered the question in the affirmative for . In Proposition 1 below we give a flexible construction that gives many pseudo-Anosov normal generators for each , thus answering Long’s question.

Penner constructed a family of pseudo-Anosov mapping classes, one for each , with the property that the stretch factors tend to 1 as tends to infinity [35]. We show in Proposition 3 that each of these small stretch factor pseudo-Anosov mapping classes is a normal generator.

Our second main theorem shows that in fact every pseudo-Anosov mapping class with sufficiently small stretch factor is a normal generator.

Theorem 2.

If a pseudo-Anosov element of has stretch factor less than then it normally generates .

For each there are pseudo-Anosov elements of that satisfy the hypothesis of Theorem 2. Indeed, for we may use the fact that there are pseudo-Anosov mapping classes with stretch factor less than (see [1, Proposition A.1]) and for we may appeal to the example given by Hironaka [16, Theorem 1.5]; see Table 1 in her paper. On the other hand, for it is known that there are no pseudo-Anosov mapping classes that satisfy the hypothesis of Theorem 2. Indeed in these cases the smallest stretch factors are known and they are greater than ; for this is classical and for this is due to Cho and Ham [5].

We can make precise one sense in which the mapping classes with stretch factor less than are abundant. A theorem of Leininger and the second author of this paper [26, Theorem 1.3] has the following consequence: for any there is a polynomial with degree and with positive leading term so that the number of pseudo-Anosov elements of with stretch factor less than is bounded below by for .

Another way to state Theorem 2 is:

If a pseudo-Anosov mapping class lies in any proper normal subgroup of , its stretch factor is greater than .

As such, Theorem 2 generalizes work of Agol, Farb, Leininger, and the second author of this paper. Farb, Leininger, and the second author proved that if a pseudo-Anosov mapping class is contained in with then the stretch factor is greater than [11, Theorem 1.7]. Agol, Leininger, and the second author proved that if a pseudo-Anosov mapping class is contained in then the stretch factor is greater than [1, Theorem 1.1]. Our Theorem 2 improves upon these results in two ways: from congruence subgroups to arbitrary normal subgroups and from 1.218 and 1.00031 to .

With regard to the first improvement, there do indeed exist examples of normal subgroups that are not contained in congruence subgroups (and hence not covered by the theorems of Agol, Farb, Leininger, and the second author). Such examples were recently constructed by Clay, Mangahas, and the second author [6], solving a problem posed in an earlier version of this paper [24, Problem 1.5].

With regard to the second improvement, we note that when using the bound , we only obtain examples of pseudo-Anosov mapping classes with stretch factor less than 1.00031 when is at least 3,106.

It is natural to ask how sharp the bound is in Theorem 2. In other words, what is the infimum of the stretch factors of all pseudo-Anosov mapping classes lying in any proper normal subgroup of any ? Farb, Leininger, and the second author [11] showed that for each there are pseudo-Anosov elements of with stretch factor at most 62. Thus the infimum lies between and 62. On the other hand, Farb, Leininger, and the second author proved that for the th term of the Johnson filtration of the smallest stretch factor tends to infinity as does (independently of ), so this gives a sequence of normal subgroups for which the bound of Theorem 2 becomes decreasingly sharp. It is an interesting problem, already raised by Farb, Leininger, and the second author [11], to understand the smallest stretch factors in various specific normal subgroups, such as .

Pseudo-Anosov elements with large stretch factor

Having given many examples of pseudo-Anosov normal generators with small stretch factor, we turn to the question of what other kinds of pseudo-Anosov normal generators may exist. In this vein, Ivanov [18, Problem 13] made the following conjecture in his 2006 problems paper:

Conjecture. If is a pseudo-Anosov element of a mapping class group with sufficiently large dilatation coefficient, then the subgroup of normally generated by is a free group having as generators the conjugates of . More cautiously, one may conjecture that the above holds for a sufficiently high power of a given pseudo-Anosov element .

(The term “dilatation coefficient” is interchangeable with the term “stretch factor.”) In Proposition 2 below we give for each a flexible construction of pseudo-Anosov normal generators for with arbitrarily large stretch factor. Since is not a free group, this in particular disproves the first part of Ivanov’s conjecture.

Much more than this, we have the following theorem. In the statement, note that is when and it is when .

Theorem 3.

Let . For each there are pseudo-Anosov mapping classes with arbitrarily large stretch factors whose normal closures in are equal to .

It is natural to ask if other normal subgroups, such as the Johnson kernel and the other terms of the Johnson filtration, can be obtained as the normal closure in of a single pseudo-Anosov mapping class.

The second part of Ivanov’s conjecture is (perhaps intentionally) ambiguous: the word “a” can be interpreted as either “some” or “any.” Dahmani, Guirardel, and Osin proved that every pseudo-Anosov mapping class has a large power (depending only on ) whose normal closure is an all pseudo-Anosov infinitely generated free group [7]. This confirms the second part of Ivanov’s conjecture with the “some” interpretation. Their theorem also answers another question in Ivanov’s problem list [18, Problem 3], which asks if there are any normal, all pseudo-Anosov subgroups of .

Our next theorem disproves the second part of Ivanov’s conjecture with the “any” interpretation. In the statement, the curve graph is the graph whose vertices are homotopy classes of simple closed curves in and whose edges are pairs of vertices with disjoint representatives in . Masur and Minsky proved that the asymptotic translation length for a pseudo-Anosov element of is a positive real number [31, Proposition 4.6].

Theorem 4.

For each there are pseudo-Anosov mapping classes with the property that all of their odd powers normally generate . Consequently, there are pseudo-Anosov mapping classes with arbitrarily large asymptotic translation lengths on that normally generate .

The pseudo-Anosov mapping classes used to prove Theorem 4 are not generic, as their invariant foliations have nontrivial symmetry groups (cf. [30]). We suspect that if the invariant foliations have no symmetries then the normal closures of all sufficiently large powers are free groups, as in Ivanov’s conjecture. We are also led to ask: if the asymptotic translation length of a pseudo-Anosov mapping class is large, can its normal closure be anything other than the mapping class group, a free group, or perhaps the extended Torelli group?

Moduli spaces

The group can be identified with the orbifold fundamental group of , the moduli space of Riemann surfaces of genus . This means that Theorems 1 and 2 can both be recast in terms of the geometry of normal covers of .

Periodic elements of correspond to orbifold points in and so Theorem 1 can be interpreted as saying that the only orbifold points in a proper normal cover of are those arising from the hyperelliptic involution. In particular, these orbifold points all have order 2 and lie along the hyperelliptic locus.

Torelli space is the normal cover of corresponding to the Torelli group. This space can be described as the space of Riemann surfaces with homology framings. A further consequence of Theorem 1 is that every normal cover of not covered by Torelli space is a manifold. In fact a normal cover of is a manifold if and only if it is not covered by the quotient of Torelli space given by the action of .

If we endow with the Teichmüller metric, then pseudo-Anosov elements of correspond exactly to geodesic loops in . The length of a geodesic loop is the logarithm of the corresponding stretch factor. Theorem 2 can thus be interpreted as saying that geodesic loops in whose lengths are less than do not lift to loops in any proper normal cover of . The existence of pseudo-Anosov normal generators for with arbitrarily large stretch factor, shown in Proposition 2, implies the existence of arbitrarily long geodesic loops in that do not lift to loops in any proper normal cover.

The well-suited curve criterion

All of the results about normal generators in this paper are derived from a simple, general principle for determining when a mapping class normally generates the mapping class group. We call this principle the well-suited curve criterion. The principle is that if we can find a curve so that the configuration is “simple” enough, then normally generates the mapping class group. We give many concrete manifestations of this principle in the paper, namely, as Lemmas 2, 3, 4, 1, 2, 1, 3, and 4 and Proposition 3.

Our first example of the well-suited curve criterion, Lemma 2 below, takes the following form for :

If lies in with and if is a nonseparating curve in with then is a normal generator for .

The well-suited curve criterion is a very general and widely applicable principle, and there are many variations besides the ones introduced in this paper. We expect that this principle can be leveraged to address other problems about mapping class groups and related groups. For example, we use a variant to prove our result about normal generators for congruence subgroups, Theorem 3 below. We also use the principle to give normal generators for certain linear groups.

While having a simple configuration for and is a powerful sufficient criterion for a mapping class to be a normal generator, it is not a necessary condition. This point is underscored by Theorem 4, which gives examples of normal generators with large translation length on the curve graph . For these examples, the distance between every curve and its image is large, which means that each curve forms a complicated configuration with its image. Still there is a way to formulate the well-suited curve criterion as a necessary, as well as sufficient, condition for a mapping class to be a normal generator. We state in Proposition 3 below, the most general version of our well-suited curve criterion, which says that a mapping class is a normal generator if and only if a certain associated curve graph is connected.

Overview of the paper

The remainder of the paper is divided into two parts. In the first part, Sections 25, we give special cases of the well-suited curve criterion and then use them to prove all of our main results aside from Theorem 2. In the second part, Sections 610, we prove Theorem 2, which is by far our most technical result.

We begin the first part of the paper by proving in Section 2 several special cases of the well-suited curve criterion. In Section 3, we use the special cases of the criterion to prove our Theorem 1 about periodic elements and to give our extension of Funar’s theorem. In Section 4 we again use the special cases of the criterion to answer Long’s question in the affirmative, to disprove Ivanov’s conjecture, and to prove Theorems 3 and 4. In Section 5 we apply our results about mapping class groups to give normal generators for certain linear groups.

The second part of the paper begins with Section 6, in which we relate small stretch factor to geometric intersection numbers for curves and lay out the plan for the proof of Theorem 2. In Section 7 we state and prove the general well-suited curve criterion, Proposition 3, and use it to prove one case of Theorem 2. Then in Section 8 we address the main technical obstacle of the paper, by giving a detailed analysis of all possible configurations of certain triples of mod 2 homologous curves that intersect in at most two points pairwise on any closed surface (there are 36 configurations up to stabilization). In Section 9 we give a variant of the well-suited curve criterion that applies to many of the configurations from Section 8. Finally, in Section 10 we use the results of Sections 69 in order to prove Theorem 2.

Acknowledgments

The authors are supported by NSF Grants DGE - 1650044 and DMS - 1510556. We would like to thank Mladen Bestvina, Lei Chen, Benson Farb, Søren Galatius, Asaf Hadari, Chris Leininger, Marissa Loving, Curtis McMullen, Gregor Masbaum, Andrew Putman, Nick Salter, Balázs Strenner, and Nick Vlamis for helpful conversations. We are also grateful to Mehdi Yazdi for bringing Long’s question to our attention at the 2017 Georgia International Topology Conference.

2. The well-suited curve criterion: special cases

As discussed in the introduction, the well-suited curve criterion is the principle that if is a mapping class and is a curve in so that the configuration is simple enough, then the normal closure of is equal to the mapping class group. We will give several examples of this phenomenon in this section, in Lemmas 2, 3, and 4. Besides serving as a warmup for the full version of the well-suited curve criterion, these special cases also suffice to prove Theorem 1, to answer Long’s question, and to resolve Ivanov’s conjecture.

Curves and intersection number

In what follows we refer to a homotopy class of essential simple closed curves in as a “curve” and we write for the geometric intersection number between curves and .

We will write for the element of represented by a curve (the ambiguity comes from the two choices of orientation). We will write for the corresponding element of . A useful fact is that whenever we have if and only if .

Finally, we write for the absolute value of the algebraic intersection number between two elements of corresponding to and . We will refer this number as simply the algebraic intersection number, as we will have no need to discuss the signed algebraic intersection number.

Normal generators for the commutator subgroup

The following lemma, along with Lemmas 2 and 3, already appears in the paper by Harvey–Korkmaz [15, Lemma 3]. The ideas also appeared in the earlier works of McCarthy–Papadopoulos [32] and Luo [29]. All of our well-suited curve criteria will be derived from this lemma.

The conclusions of Lemmas 2 and 3 only give that the normal closure of a given element contains the commutator subgroup of . For it is well-known that is perfect [12, Theorem 5.2] and so in these cases the lemmas imply that is a normal generator.

Lemma 1.

Suppose and are nonseparating curves in with . Then the normal closure of is equal to the commutator subgroup of .

Proof.

We will show the two inclusions in turn. For , it is known that is generated by Dehn twists about nonseparating curves. It is also known that the abelianization of is cyclic. Since the Dehn twists about any two nonseparating curves are conjugate it follows that , and hence its normal closure, lies in the commutator subgroup of .

It remains to show that the commutator subgroup is contained in the normal closure of . Lickorish proved that there is a generating set for where the generators are Dehn twists about nonseparating curves in and where each is at most 1 [12, Theorem 4.13]. The commutator subgroup of is thus normally generated by the various . When , the corresponding commutator is trivial. The nontrivial commutators are all conjugate in to , where and are the curves in the statement of the lemma. Therefore it suffices to show that the single commutator is contained in the normal closure of .

It is a general fact that if and are elements of a group and is a normal subgroup of containing then is contained in . Indeed, if we consider the quotient homomorphism then and map to the same element, and so maps to the identity. Applying this general fact to our situation, we have that is contained in the normal closure of , as desired. ∎

Two well-suited curve criteria for nonseparating curves

The next two lemmas are special cases of the well-suited curve criterion.

Lemma 2.

Let and let . Suppose that there is a nonseparating curve in so that . Then the normal closure of contains the commutator subgroup of .

Proof.

Consider the commutator . Since this commutator is equal to the product of and , it lies in the normal closure of . Since is equal to the commutator is also equal to . Since , the lemma now follows from Lemma 1. ∎

Figure 2. The curves , , and in the proof of Lemma 3
Lemma 3.

Let and let . Suppose that there is a nonseparating curve in so that and . Then the normal closure of contains the commutator subgroup of .

Proof.

It follows from the hypotheses on and that there is an additional curve with , , and ; see Figure 2. The commutator again lies in the normal closure of . Since acts transitively on pairs of disjoint, non-homologous curves in , there is an taking the pair to the pair . The conjugate , which also lies in the normal closure of , is equal to . Thus the product

lies in the normal closure of . An application Lemma 2 completes the proof. ∎

Figure 3. The configurations where is separating and

A well-suited curve criterion for separating curves

We will now use Lemma 3 to obtain an instance of the well-suited curve criterion for separating curves, as follows.

Lemma 4.

Let and let . Suppose that there is a separating curve in with . Then the normal closure of contains the commutator subgroup of .

Proof.

Let be a separating curve in with . Since the intersection number between two separating curves is even, we have that is either 0 or 2. In each case there is only one possible configuration up to homeomorphism and the genera of the complementary regions; see Figure 3.

In each case we may find nonseparating curves and so that and lie on different sides of and on the same side of . Since and lie on different sides of , it follows that and lie on different sides of . Therefore it must be that either and lie on different sides of or and do (or both). Without loss of generality, suppose and lie on different sides of . Then clearly and . An application of Lemma 3 completes the proof. ∎

3. Application: periodic elements

In this section we apply the special cases of the well-suited curve criterion from Section 2 to determine the normal closure of each periodic element of each . The main technical result is the following.

Proposition 1.

Let and let be a nontrivial periodic element of that is not a hyperelliptic involution. Then the normal closure of contains the commutator subgroup of .

Since is perfect when , Proposition 1 immediately implies Theorem 1. Later in the section we will use Proposition 1 to determine the normal closures of all periodic elements of and . At the end of the section, we prove Corollary 5, which gives a condition on so that the th power subgroup of is the whole group.

Our proof of Proposition 1 requires a lemma about roots of the hyperelliptic involution, Lemma 2 below. Before giving this lemma, we begin with some preliminaries.

Standard representatives

It is a classical theorem of Fenchel and Nielsen [12, Theorem 7.1] that a periodic mapping class is represented by a homeomorphism whose order is equal to that of . Moreover, is unique up to conjugacy in the group of homeomorphisms of . We refer to any such as a standard representative of .

The Birman–Hilden theorem

We now recall a theorem of Birman and Hilden. Let be a hyperelliptic involution of (a homeomorphism as in Figure 1) and let be the resulting hyperelliptic involution in . Birman and Hilden proved [2, Theorem 1] that for there is a short exact sequence

where is the centralizer in of and is a sphere with marked points. The map is defined as follows: it is proved [3, Theorem 4] that each element of has a representative that commutes with , and so can be pushed down to a homeomorphism of the quotient , which is a sphere with marked points, namely, the images of the fixed points of . We note that the above exact sequence is not correct as stated for , as the given map is not well-defined.

Roots of the hyperelliptic involution

The next lemma describes a property of roots of the hyperelliptic involution that will be used in the proof of Proposition 1.

Lemma 2.

Let . Suppose is an th root of a hyperelliptic involution (with ). Then there is a power of that is not the identity or the hyperelliptic involution and that has a standard representative with a fixed point.

Proof.

We first dispense with the case . In this case all periodic elements are given by rotations of either the hexagon or square. In particular, they all have fixed points. Any nontrivial root of the hyperelliptic involution is a periodic element, and so the first power of satisfies the conclusion.

Now assume that . Fix a hyperelliptic involution and corresponding mapping class as above. Suppose that is the th root of . It follows that lies in , the centralizer of . If is the map from the aforementioned exact sequence of Birman and Hilden, then is a periodic element of .

Let be a standard representative of . By ignoring the marked points, we may regard as a finite-order homeomorphism of . A theorem of Brouwer, Eilenberg, and de Kerékjártó [4, 8, 10] states that every finite-order orientation-preserving homeomorphism of is conjugate to a rotation. In particular, has two fixed points in .

There are two homeomorphisms of that are lifts of ; they differ by the hyperelliptic involution . One of these lifts is a standard representative for and one is a standard representative for . If at least one of the fixed points of is marked point, then both lifts of to have a fixed point. In particular has a fixed point, as desired.

Now suppose that neither of the fixed points of are marked points. In this case we will show that satisfies the conclusion of the lemma.

Let be one of these fixed points and let be one point of the preimage in . One of the lifts of fixes and one interchanges it with . The one that fixes cannot be a representative of since no power of this homeomorphism is (the only fixed points of are the preimages of the marked points in ). So the lift of that interchanges and is a standard representative of .

The homeomorphism fixes . Clearly then is also not equal to or the identity, since does not fix . Thus is not the identity or a hyperelliptic involution and its standard representative fixes a point in , as desired. ∎

Figure 4. A cyclic cover of surfaces

Proof of the theorem

We now prove Proposition 1, which, as described above, implies Theorem 1.

Proof of Proposition 1.

Let be a nontrivial periodic mapping class and assume that is not a hyperelliptic involution. Let be a standard representative of and let denote the cyclic group of homeomorphisms of generated by .

We treat separately three cases:

Case 1. the action of is free,

Case 2. the action of is not free and has order 2, and

Case 3. the action of is not free and has order greater than 2.

We begin with the first case. If the action of is free then this action is a covering space action. Every cyclic covering map is equivalent to one of the covering maps indicated in Figure 4. In particular, we can find a curve so that and satisfy the hypotheses of Lemma 3. It follows from Lemma 3 that the normal closure of contains the commutator subgroup of , as desired.

Figure 5. Rotation by gives a mapping class of order 2

We now treat the second case. There is a classification of homeomorphisms of of order 2 that goes back to the work of Klein [20]; see Dugger’s paper [9] for a modern treatment. In the cases where there is a fixed point in , such a homeomorphism is conjugate to one of the ones indicated by Figure 5; there is a 1-parameter family, according to the number of handles above the axis of rotation. In particular, the conjugacy class of one of these homeomorphisms is completely determined by the genus of the quotient surface . When the genus of the quotient is 0, the homeomorphism is a hyperelliptic involution, which is ruled out by hypothesis. When the genus of the quotient is positive, we can again find a curve as in Lemma 3. Again by Lemma 3 the normal closure of contains the commutator subgroup of .

Finally we treat the third case, where the order of is greater than 2 and the action of on is not free. Since the action of on is not free, some power of has a fixed point. We may choose this power so that has a fixed point and is not a hyperelliptic involution or the identity. This is obvious if is not a root of a hyperelliptic involution and it follows from Lemma 2 otherwise. Without loss of generality we may replace by this power with a fixed point, since the normal closure of a power of is contained in the normal closure of ; we may continue to assume that the order of the new is greater than two, for if not we may apply Cases 1 and 2 above.

A theorem of Kulkarni [22, Theorem 2] states that if is a finite order homeomorphism of that has a fixed point, then there is a way to represent as a quotient space of some regular -gon in such a way that is given as rotation of the polygon by some multiple of . We apply this theorem to . Let be the resulting -gon (so is obtained from by identifying the sides of in pairs in some way).

Let be a line segment in that connects the midpoints of two edges that are identified in . Then represents a curve in . We may assume that represents a nontrivial curve in , for if all curves in coming from these line segments were trivial, then we would have , in which case is trivial.

We may regard as a rotation of by some multiple of . The image of the line segment under is another line segment . Since has order greater than 2, it follows that is not equal to , and hence the number of intersections between these line segments is either 0 or 1. Moreover, this intersection number is equal to (here we are regarding as a curve in ).

If is a separating curve in , then the proposition follows from an application of the well-suited curve criterion for separating curves (Lemma 4). We may henceforth assume that is nonseparating.

If , then it follows from Lemma 2 that the normal closure of contains the commutator subgroup of , as desired. Finally suppose that . We would like to show that . To do this, it is enough to show that , , and are not all equal (if then it would follow that ).

Figure 6. Three disjoint homologous curves in

Since the order of is greater than 2 the line segments , , and are all distinct. If we may again Lemma 2 to to conclude that the normal closure of , hence contains the commutator subgroup. So we may assume that , , and are all disjoint. If we cut along the three curves corresponding to , , and then there is a region bordering all three curves, namely, the region of containing the image of the center of . On the other hand, if we have three disjoint curves in that are all homologous, then they must separate into three components, and each region abuts two of the three curves (refer to Figure 6). Thus, , , and are not all equal and we are done. ∎

Normal closure of the hyperelliptic involution

To complete the classification of normal closures of periodic elements in it remains to deal with the case of the hyperelliptic involution and also the cases of and . The following proposition already appears in the paper by Harvey and Korkmaz [15, Theorem 4(b)]. In the statement, let denote the standard symplectic representation .

\labellist\hair

2pt \pinlabel [ ] at 135 27 \pinlabel [ ] at 135 90 \endlabellist

Figure 7. The normal closure of in is
Proposition 3.

Let and let be a hyperelliptic involution. Then the normal closure of in is the preimage of under .

Proof.

The image of under is . Since the latter is central in it follows that the normal closure of is contained in the preimage of . It remains to show that the normal closure of contains the Torelli group .

It is a theorem of D. Johnson [19, Theorem 2] that is equal to the normal closure in of the mapping class , where and are the curves in indicated in Figure 7 (this mapping class is called a bounding pair map of genus 1). Since we have that

Thus is equal to a product of two conjugates of . In particular it is contained in the normal closure of . By Johnson’s result, the normal closure of contains and we are done. ∎

Low genus cases

We now discuss the cases of and . In both cases the hyperelliptic involution is unique and is central in . Thus the normal closure of in both cases is the cyclic group of order 2 generated by . By Proposition 1 the normal closure of any other periodic element of contains the commutator subgroup, and hence the normal closure of an element is completely determined by the image of in the abelianization of .

The abelianizations of and are isomorphic to and , respectively; see [12, Section 5.1.3]. The image of any Dehn twist about a nonseparating curve in either case is equal to 1. Therefore, if is a periodic element of that is not a hyperelliptic involution and is equal to a product of Dehn twists about nonseparating curves, then the normal closure of in is equal to the preimage under the abelianization map of the group generated by (here is 12 when and 10 when ). In particular if we consider any periodic element that is not a hyperelliptic involution then the normal closure has finite index in . For a complete list of periodic elements in and and realizations of these elements as products of Dehn twists about nonseparating curves, see the paper of Hirose [17, Theorem 3.2] (we note that the second in Hirose’s theorem should be ). In each case the image of the periodic element is not a generator for the abelianization. So for equal to 1 or 2, the normal closure of any periodic element is a proper subgroup of .

Finite generating sets

We already mentioned results of Harvey, Korkmaz, McCarthy, Papadopoulos, Yoshihara, and the first author of this paper that give periodic normal generators for the mapping class group. Much more than this, Korkmaz proved that only two conjugate elements of order are needed to generate . Similarly, Monden proved that three conjugate elements of order 3 are needed, and only four conjugate elements of order 4 are needed. Yoshihara proved that three conjugate elements of order 6 are required for . Finally the first author proved that for and only three conjugate elements of order are needed, and for and only four conjugate elements are needed. Based on these results and Theorem 1 we are led to the following question.

Question 4.

Is there a number , independent of , so that if is a periodic normal generator of then is generated by conjugates of ?

We emphasize that in the question is independent of both and .

Power subgroups

As in the introduction, let denote the least common multiple of the orders of the periodic elements of . We have the following corollary of Theorem 1.

Corollary 5.

Let . Suppose that is not divisible by . Then the subgroup of generated by the th powers of all elements is equal to the whole group .

This result improves on a theorem of Funar [14, Theorem 1.16(2)] who showed the analogous result with replaced by . Note that is a factor of since has 4 and as factors. Thus our result indeed recovers the theorem of Funar. For comparison, when the number is 360 and is 18.

Proof of Corollary 5.

Since does not divide , then for some prime we have that has a factor whereas only has a factor with .

If this is an odd prime, then contains a periodic element whose order has as a factor. The element is nontrivial and its order has as a factor. Since is odd it follows that is not a hyperelliptic involution. By Theorem 1, is a normal generator for . Since is contained in the th power subgroup we are done in this case.

If instead is 2, then has a factor whereas only has a factor with . But then has a factor . This implies that contains a periodic element whose order has as a factor, and we have that is nontrivial and its order has as a factor, which is at least 4. Hence is nontrivial and not the hyperelliptic involution. Again an application of Theorem 1 completes the proof. ∎

If we only consider periodic elements when analyzing power subgroups, we could at most hope to replace by in the corollary (as the th power of every periodic element is trivial). However, for some the analysis fails with this replacement. For instance, when is a power of 2, the element of order is the only element that is nontrivial when raised to the power , and this power is the hyperelliptic involution.

Of course there do exist values of so that the th power subgroup of is not . Indeed for any group with a proper subgroup of finite index, there is an so that the th power group of is a subgroup of and hence is not equal to .

4. Application: Long’s question and Ivanov’s conjecture

In this section we construct, for each , a pseudo-Anosov mapping class that normally generates (for pseudo-Anosov mapping classes are usually called Anosov, but we will not make this distinction). As discussed in the introduction, it follows from our Theorem 2 that such mapping classes exist when . The examples in this section are simple and explicit and only use the special cases of the well-suited curve criterion given in Section 2, whereas the proof of Theorem 2 is more involved and requires the more general version of the well-suited curve criterion given in Section 7.

We begin with a very simple family of pseudo-Anosov mapping classes that normally generate, then we give examples that have large stretch factor, examples that have small stretch factor, and examples that have large translation length on the curve graph. Finally we give examples with large stretch factor with normal closure equal to any given congruence subgroup. As described in the introduction, the examples in this section answer Long’s question in the affirmative and resolve Ivanov’s conjecture in the negative.

\labellist\hair

2pt \pinlabel [ ] at 83 112 \pinlabel [ ] at 193 112 \pinlabel [ ] at 303 112 \pinlabel [ ] at 28 53 \pinlabel [ ] at 138 53 \pinlabel [ ] at 248 53 \pinlabel [ ] at 358 53 \endlabellist

Figure 8. The and used in the definition of and

First examples

For each , let and where the and are the curves in indicated in Figure 8. The figure shows the case but there is an obvious generalization for all other ; when the curves and are parallel.

Consider the mapping class (note depends on ). The following proposition answers in the affirmative the question of Long from the introduction.

Proposition 1.

For each the mapping class is pseudo-Anosov and it normally generates .

Proof.

There are well-known constructions of pseudo-Anosov mapping classes due to Thurston and to Penner where certain products of Dehn twists are shown to be pseudo-Anosov; see [12, Theorems 14.1 and 14.4]. The product is both an example of the Thurston construction and an example of the Penner construction. In particular, is pseudo-Anosov for all .

Consider the action of on . Since is equal to 0 for all and is equal to 0 for we have that . It follows that is equal to 1. Thus by Lemma 2 the normal closure of contains the commutator subgroup of . For it follows that normally generates . For equal to 1 or 2, it is enough now to observe that since the sum of the exponents of the Dehn twists in the definition of is 1. As such the image of in the abelianization of is 1, and hence a generator for the abelianization. Thus in these cases is a normal generator as well. ∎

Examples with large stretch factor

There is great flexibility in the construction of . Indeed, we can alter any of the exponents, except on , and we obtain another normal generator for when . If we also preserve the condition that the exponent sum is relatively prime to 12 or 10, we obtain a normal generator for or , respectively. We take advantage of this flexibility to prove the following.

Proposition 2.

For each there are pseudo-Anosov mapping classes with arbitrarily large stretch factors that normally generate .

Proof.

Fix some and consider again the mapping classes and as above. Much more than proving is pseudo-Anosov, Thurston proved that there is a positive number and a homomorphism

with

and so that is pseudo-Anosov if and only if its image in (or, rather, a representative in ) has an eigenvalue ; moreover in this case is the stretch factor of (the number is the Perron–Frobenius eigenvalue of , where is the intersection matrix between the and ; see [12]).

Consider then the mapping class . Its image in is

The trace of this matrix is larger than 2, and so it has two real eigenvalues. The larger of these eigenvalues is strictly larger than the trace, which is . In particular, each mapping class is pseudo-Anosov, and the stretch factors tend to infinity as tends to infinity. As in the proof of Proposition 1, we have and so by Lemma 2 the normal closure of each of these mapping classes contains the commutator subgroup of . This completes the proof for .

For the image of in the abelianization of is . For each equal to 1, 3, 5, or 7 mod 10 the image of is therefore relatively prime to 10 and so the proposition is proven for . For the image of is , which is relatively prime to 12 for equal to 0, 1, 3, or 4 mod 6. ∎

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Figure 9. The curves and the rotation used in Penner’s construction of pseudo-Anosov mapping classes with small stretch factor

Examples with small stretch factor

There are many other explicit examples of pseudo-Anosov mapping classes that satisfy the special cases of the well-suited curve criterion given in Section 2 and hence normally generate . Here we discuss one famous family of examples due to Penner.

Consider the curves , , and in indicated in Figure 9 and consider the order rotation of indicated in the same figure. Penner proved that the product is pseudo-Anosov (to prove this he notes that the th power is an example of his Dehn twist construction for pseudo-Anosov mapping classes). By inspection we observe that Penner’s mapping class, together with the curve in Figure 9, satisfies Lemma 3. We thus have the following proposition.

Proposition 3.

For each , Penner’s mapping class is a normal generator for .

For the mapping class maps to 5 in and so Penner’s mapping class maps to 6. Thus the image of Penner’s mapping class has index 2 in . On the other hand, is pseudo-Anosov and normally generates.

Penner proved that the stretch factors of his mapping classes are bounded above by for all . Hence they form a sequence of pseudo-Anosov mapping classes whose stretch factors tend to 1 as tends to infinity. As such, these examples served as the initial inspiration for our Theorem 2, which says that all pseudo-Anosov mapping classes with sufficiently small stretch factor are normal generators.

Smallest stretch factors

For it is an open problem to determine which mapping classes realize the smallest stretch factor. Nevertheless, we do know in these cases that the smallest stretch factors are less than and so by Theorem 2 the minimizers are normal generators.

For equal to 1 or 2, we know precisely which mapping classes realize the smallest stretch factor in . For the minimum is given uniquely by the conjugacy class of where and are as shown in Figure 8. The stretch factor of is . The mapping class is not a normal generator since the image of in , the abelianization of , is 0. On the other hand, the normal closure of does contain the commutator subgroup of . This is because and so satisfies Lemma 2.

For , Cho and Ham [5] proved that the minimum stretch factor is the largest real root of , which is approximately 1.72208. Lanneau and Thiffeault [25] gave an independent proof and also classified the conjugacy classes of all of the minimizers and gave explicit representatives of these conjugacy classes in terms of Dehn twists. Again using the labels in Figure 8, these are:

The images of these mapping classes in the abelianization are 4 and 0, respectively, and so neither is a normal generator. As in the case the normal closures of both and contain the commutator subgroup of since and .

Examples with large translation length

In this section, we have already given examples of pseudo-Anosov normal generators for with either small or large stretch factor. All of these examples have translation length in at most 2 since they each satisfy one of the well-suited curve criteria given in Lemmas 2 and 3. It follows that the asymptotic translation lengths are also bounded above by 2.

We will now prove Theorem 4, which states that for each there are pseudo-Anosov mapping classes with the property that all of their odd powers are normal generators. Since the asymptotic translation length on the curve graph is multiplicative, and since the asymptotic translation length of every pseudo-Anosov mapping class is positive, it follows that there are pseudo-Anosov normal generators with arbitrarily large asymptotic translation lengths. Because large translation length implies large stretch factor, Theorem 4 implies Proposition 2 in the cases when .

Proof of Theorem 4.

Fix . Let denote the dihedral group of order . There is a standard action of on by orientation-preserving homeomorphisms. In this way, we identify as a subgroup of . We denote the quotient by . As an orbifold, is a sphere with five orbifold points.

Consider any pseudo-Anosov homeomorphism of , thought of as a sphere with five marked points. Up to taking a power, we may assume that this homeomorphism lifts to a pseudo-Anosov homeomorphism of ; see [12, Section 14.1.1].

The homeomorphism lies in the normalizer of , and so we may identify with an element of the automorphism group of ; it acts by conjugation. Let be the element of corresponding to rotation by under the standard action of on the -gon. Since and are the only elements of that are conjugate in , it must be that the automorphism of induced by maps to either or . Let correspond to some reflection of the -gon. Up to replacing by we may assume that the automorphism of induced by maps to . Since fixes the invariant foliations for and does not change the stretch factor, the new is a pseudo-Anosov element of . If we identify as an element of then . By construction for all odd .

Consider the mapping class for odd. We have

Thus lies in the normal closure of in . By the well-suited curve criterion of Lemma 3 (and using the assumption ) the mapping class , hence , is a normal generator for . ∎

Torelli groups and congruence subgroups

Having found an abundance of pseudo-Anosov normal generators for the mapping class group, we now consider the question of which proper normal subgroups of arise as the normal closure of a single pseudo-Anosov mapping class.

Specifically, our next goal is to prove Theorem 3, which states that all level congruence subgroups arise as the normal closure of a pseudo-Anosov mapping class, and moreover that this mapping class can be chosen to have arbitrarily large stretch factor. The theorem covers the Torelli group as the case where . The construction in the proof is a variant of a construction of Leininger and the second author [26, Proof of Proposition 5.1].

The proof of Theorem 3 requires the following technical lemma, whose statement and proof are well known to experts, but are not easily found in the literature.

Lemma 4.

Let and be two multicurves that fill . The stretch factors of the mapping classes

tend to infinity as does.

Since the all commute with each other, since the all commute with each other, and since the stretch factor of a mapping class is the same as that of its inverse, the lemma holds equally well if we attach the exponent to any or attach the exponent to any .

Proof of Lemma 4.

Consider the intersection graph with vertices and edges for each pair (this is really a multigraph). Since and fill , this graph is connected. Let be half the maximum path-length distance between two -vertices of .

Let denote the multicurve obtained from by replacing with parallel copies of , denoted . Let and be the products of the Dehn twists about the curves in and , respectively. We thus have

Let be the intersection graph for and . This graph has vertices and is the blowup of obtained by replacing the vertex with vertices .

Let be the matrix whose rows correspond to the curves of , whose columns correspond to the curves of , and whose entries are given by the corresponding intersection numbers. The matrix is a square matrix with rows and columns corresponding to the curves of . Each entry is the number of paths of length 2 between the corresponding vertices of . Similarly, the entries of any power