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Injective homomorphisms of mapping class groups of non-orientable surfaces

Abstract

Let be a compact, connected, non-orientable surface of genus with boundary components, with and , and let be the mapping class group of . We show that, if is a finite index subgroup of and is an injective homomorphism, then there exists such that for all . We deduce that the abstract commensurator of coincides with .

Injective homomorphisms of mapping class groups of non-orientable surfaces]Injective homomorphisms of mapping class groups of non-orientable surfaces E Irmak]Elmas Irmak \givennameElmas \surnameIrmak \urladdr L Paris]Luis Paris \givennameLuis \surnameParis \urladdr \subjectprimarymsc200057N05 \volumenumber \issuenumber \publicationyear \papernumber \startpage \endpage \doi \MR \Zbl \published \publishedonline \proposed \seconded \corresponding \editor \version \theoremstyledefinition \numberwithinequationsection

1 Introduction

Let be a connected and compact surface that can be non-orientable and have boundary but whose Euler characteristic is negative. We denote by the group of homeomorphisms of . The mapping class group of , denoted by , is the group of isotopy classes of elements of . Note that it is usually assumed that the elements of preserve the orientation when is orientable. However, the present paper deals with non-orientable surfaces and we want to keep the same definition for all surfaces, so, for us, the elements of can reverse the orientation even if is orientable. On the other hand, we do not assume that the elements of pointwise fix the boundary. In particular, a Dehn twist along a circle isotopic to a boundary component is trivial. If is not connected, the mapping class group of , denoted by , has the same definition. Although our results concern connected surfaces, we shall use in some places mapping class groups of non-connected surfaces.

The purpose of this paper is to prove the following.

Theorem 1.1

Let be a compact, connected, non-orientable surface of genus with boundary components, with and , let be a finite index subgroup of , and let be an injective homomorphism. Then there exists such that for all .

The same result is known for orientable surfaces (see Irmak [9, 10, 11], Behrstock–Margalit [4], Bell–Margalit [5]). A first consequence of Theorem 1.1 is as follows. This corollary was proved in Atalan–Szepietowski [1, 3] for compact, connected, non-orientable surfaces of genus . Our contribution is just an extension to compact, connected, non-orientable surfaces of genus and .

Corollary 1.2

Let be a compact, connected, non-orientable surface of genus with boundary components, with and . Then and .

Note that the ideas of Atalan–Szepietowski [1, 3] cannot be extended to the more flexible framework of Theorem 1.1 because they use rather precise relations in that do not necessarily hold in a finite index subgroup.

Let be a group. We denote by the set of triples , where and are finite index subgroups of and is an isomorphism. Let be the equivalence relation on defined as follows. We set if there exists a finite index subgroup of such that for all . The (abstract) commensurator of is defined to be the quotient . This is a group. A more general consequence of Theorem 1.1 is the following.

Corollary 1.3

Let be a compact, connected, non-orientable surface of genus with boundary components, with and . Then .

Let be a compact, connected, non-orientable surface of genus with boundary components, with and . We denote by the complex of curves of and by the subcomplex of formed by the isotopy classes of two-sided circles. For we denote by the intersection index of and and, for , we denote the Dehn twist along . A map is called a super-injective simplicial map if the condition is equivalent to the condition for all .

The proof of Theorem 1.1 is based on two other theorems. The first one, proved in Irmak–Paris [13, Theorem 1.1], says that, if is a super-injective simplicial map, then there exists such that for all . Let be the subfamily of formed by the isotopy classes of circles that do not bound one-holed Klein bottles. The second theorem is the aim of Section 3 of this paper (see Theorem 3.1). It says that, if is a finite index subgroup of and is an injective homomorphism, then there exists a super-injective simplicial map which sends a non-trivial power of to a non-trivial power of for all .

The proof of Theorem 3.1 uses several results on reduction classes of elements of and on maximal abelian subgroups of . These results are classical and widely used in the theory of mapping class groups of orientable surfaces, but they are more or less incomplete in the literature for mapping class groups of non-orientable surfaces. So, the aim of Section 2 is to recall and complete what is known on non-orientable case.

In order to prove Theorem 3.1 we first prove an algebraic characterization of some Dehn twists up to roots and powers (see Proposition 3.3). This characterization seems interesting by itself and we believe that it could be used for other purposes. Note also that this characterization is independent from related works of Atalan [1, 2] and Atalan–Szepietowski [3].

The final proofs of Theorem 1.1 and Corollary 1.3 are given in Section 4. The proof of Corollary 1.2 is left to the reader.

2 Preliminaries

2.1 Canonical reduction systems

From now on denotes compact, connected, non-orientable surface of genus with boundary components. A circle in is an embedding . This is assumed to be non-oriented. It is called generic if it does not bound a disk, it does not bound a Möbius band, and it is not isotopic to a boundary component. The isotopy class of a circle is denoted by . A circle is called two-sided (resp. one-sided) if a regular neighborhood of is an annulus (resp. a Möbius band). We denote by the set of isotopy classes of generic circles, and by the subset of of isotopy classes of generic two-sided circles.

The intersection index of two classes is . The set is endowed with a structure of simplicial complex, where a finite set is a simplex if for all . This simplicial complex is called the curve complex of . Note that acts naturally on , and this action is a simplicial action. Note also that is invariant under this action.

Let be a simplex of . Let denote the stabilizer of in . Choose pairwise disjoint representatives , , and denote by the natural compactification of . The groups and are linked by a homomorphism , called the reduction homomorphism along , defined as follows. Let . Choose a representative of such that . The restriction of to extends in a unique way to a homeomorphism . Then is the element of represented by .

We denote by the Dehn twist along a class . This is defined up to a power of , since its definition depends on the choice of an orientation in a regular neighborhhood of a representative of , but this lack of precision will not affect the rest of the paper. If is a simplex of , we set , and we denote by the subgroup of generated by . The following is a classical result for mapping class groups of orientable surfaces. It is due to Stukow [18, 19] for non-orientable surfaces.

Proposition 2.1 (Stukow [18, 19])

Let be a simplex of . Then is the kernel of and it is a free abelian group of rank .

Let . We say that is pseudo-Anosov if and for all and all . We say that is periodic if it is of finite order. We say that is reducible if there is a non-empty simplex of such that . In that case, is called a reduction system for and an element of is called a reduction class for . An element can be both periodic and reducible, but it cannot be both pseudo-Anosov and periodic, and it cannot be both pseudo-Anosov and reducible.

Let be a reducible element and let be a reduction system for . Let be the connected components of . Choose such that for all and denote by the restriction of to . We say that is an adequate reduction system for if, for all , is either periodic or pseudo-Anosov. This definition does not depend on the choice of . The following was proved by Thurston [8] for orientable surfaces and then was extended to non-orientable surfaces by Wu [20].

Theorem 2.2 (Thurston [8], Wu [20])

A mapping class is either pseudo-Anosov, periodic or reducible. Moreover, if it is reducible, then it admits an adequate reduction system.

Let . A reduction class for is an essential reduction class for if for each such that and for each integer , the classes and are distinct. We denote by the set of essential reduction classes for . Note that if and only if is either pseudo-Anosov or periodic. The following was proved by Birman–Lubotzky–McCarthy [6] for orientable surfaces and by Wu [20] and Kuno [16] for non-orientable surfaces.

Theorem 2.3 (Wu [20], Kuno [16])

Let be a non-periodic reducible mapping class. Then , is an adequate reduction system for , and every adequate reduction system for contains . In other words, is the unique minimal adequate reduction system for .

Remark

The proof of Theorem 2.3 for orientable surfaces given in Birman–Lubotzky–McCarthy [6] does not extend to non-orientable surfaces, since Lemma 2.4 in [6] is false for non-orientable surfaces and this lemma is crucial in the proof. The proof for non-orientable surfaces is partially made in Wu [20] using oriented double coverings, and it is easy to get the whole result by these means. Kuno’s approach [16] is different in the sense that she replaces Lemma 2.4 of [6] by another lemma of the same type.

If is a non-periodic reducible mapping class, then the set is called the essential reduction system for . Recall that, if is either periodic or pseudo-Anosov, then . The following lemma follows from the definition of .

Lemma 2.4

Let . Then

  • for all ,

  • for all .

Moreover, we will often use the following.

Lemma 2.5

Let be a simplex of , and, for every , let . Set . Then .

{proof}

We have for all , hence is a reduction system for . The reduction of along is the identity, hence is an adequate reduction system. Let . Set . Note that is also a reduction system for . We choose disjoint representatives , , we denote by the natural compactification of , and we denote by the component of that contains . The restriction of to is , which is neither periodic, nor pseudo-Anosov. Hence, is not an adequate reduction system. So, is the minimal adequate reduction system for , that is, .

2.2 Abelian subgroups

Let be a simplex of and let be an abelian subgroup of . We say that is a reduction system for if is a reduction system for every element of . Similarly, we say that is an adequate reduction system for if is an adequate reduction system for every element of . On the other hand, we set . The following can be proved exactly in the same way as Lemma 3.1 in Birman–Lubotzky–McCarthy [6].

Lemma 2.6

Let be an abelian subgroup of . Then is an adequate reduction system for and every adequate reduction system for contains .

The rank of is defined by if is even and by if is odd. Let be a simplex of . As ever, we set . For , we denote by the subgroup of generated by . By Proposition 2.1, is a free abelian group of rank . The rank of an abelian group is denoted by .

Kuno [16] proved that the maximal rank of an abelian subgroup of is precisely . Her proof is largely inspired by Birman–Lubotzky–McCarthy [6]. In the present paper we need the following more precise statement whose proof is also largely inspired by Birman–Lubotzky–McCarthy [6].

Proposition 2.7

Let be an abelian subgroup of . Then . Moreover, if , then the following claims hold.

  • There exists a simplex in such that and .

  • There exists such that .

  • None of the connected components of is homeomorphic to .

Remark

There are classes of circles that are not included in any simplex of of cardinality . These classes will be considered in Section 3. In particular, Part (1) of the above proposition is not immediate.

The rest of this subsection is dedicated to the proof of Proposition 2.7.

Let be a simplex of . We say that is a pants decomposition if is a disjoint union of pairs of pants. It is easily seen that any simplex of is included in a pants decomposition. The following lemma is partially proved in Irmak [12, Lemma 2.2].

Lemma 2.8

Let be a simplex of . Set and . Then

Moreover, equality holds if and only if is a pants decomposition.

{proof}

We choose a pants decomposition which contains . Clearly, and . Let be the number of connected components of . Since every connected component of is a pair of pants, we have and . These equalities imply that . It follows that , and equality holds if and only if .

Corollary 2.9

If is a simplex of , then .

{proof}

We have and . By Lemma 2.8, it follows that . If is even, then the right hand side of this inequality is , hence . If is odd, then the right hand side of this inequality is not an integer, but the greatest integer less than the right hand side is . So, we have in this case, too.

Remark

There are simplices in of cardinality . Their construction is left to the reader.

For , we denote by the orientable surface of genus and boundary components. The first part of the following lemma is well-known. The second part is proved in Scharlemann [17] and Stuko [18]. The third part is easy to prove.

Lemma 2.10
  • The mapping class groups and are finite.

  • The set is reduced to a unique element, , and , where the copy of is generated by .

  • If is a connected surface, with negative Euler characteristic, different from and , then is non-empty.

The following lemma is well-known for orientable surfaces (see Fathi–Laudenbach–Poénaru [8, Thm. III, Exp. 12] and Ivanov [14, Lem. 8.13]). It can be easily proved for a non-orientable surface from the fact the the lift of a pseudo-Anosov element of to the mapping class group of the orientable double-covering of is a pseudo-Anosov element (see Wu [20]).

Lemma 2.11

Let be a connected surface (orientable or non-orientable), and let be a pseudo-Anosov element. Then the centralizer of in is virtually cyclic.

{proof}

[Proof of Proposition 2.7] Let be an abelian subgroup of . Set and . Denote by the reduction homomorphism along . Note that since is a reduction system for . By Proposition 2.1 we have the following short exact sequence

hence . Moreover, again by Proposition 2.1, .

We denote by the set of connected components of , by the group of permutations of , and by the natural homomorphism. Note that since is of finite index in . Let and let be a connected component of . Let such that . The set is an adequate reduction system for , hence the restriction of to is either pseudo-Anosov or periodic. Moreover, by Lemma 2.10, does not contain any pseudo-Anosov element if is homeomorphic to or , and, by Lemma 2.11, the centralizer of a pseudo-Anosov element of is virtually cyclic. Let be the connected components of that are not homeomorphic to or . Then, by the above, .

Let . By Lemma 2.10, we have , hence we can choose a class . We set and . Then is a simplex of , hence, by Corollary 2.9, we have , and therefore

Assume that . Then and . Moreover, is of finite index in , hence there exists such that . Finally, for every , there exists such that the restriction of to is pseudo-Anosov. By Lemma 2.10, this implies that cannot be homeomorphic to .

3 Super-injective simplicial maps

Let be a subset of . A map is called a super-injective simplicial map if the condition is equivalent to the condition for all . It is shown in Irmak–Paris [13, Lemma 2.2] that a super-injective simplicial map on is always injective. We will see that the same is true for the set defined below (see Lemma 3.6).

Let . We say that bounds a Klein bottle if has two connected components and one of them, , is a one-holed Klein bottle. Recall that is a singleton (see Lemma 2.10 (2)). The class is then called the interior class of . The aim of this section is to prove the following theorem. This together with Theorem 1.1 of Irmak–Paris [13] are the main ingredients for the proof of Theorem 1.1.

Theorem 3.1

Let be a finite index subgroup of and let be an injective homomorphism. There exists a super-injective simplicial map satisfying the following properties.

  • Let which does not bound a Klein bottle. There exist such that and .

  • Let which bounds a Klein bottle and let be its interior class. The class bounds a Klein bottle, is the interior class of , and there exist and such that and .

Remark

It will follow from Theorem 1.1 that in Part (b) of the above theorem.

We say that a class is separating (resp. non-separating) if has two connected components (resp. has one connected component). We denote by the set of separating classes such that both components of are non-orientable of odd genus. We denote by the set of separating classes that bound Klein bottles. And we denote by the complement of in .

The rest of the section is devoted to the proof of Theorem 3.1. In Subsection 3.1 we prove the restriction of Theorem 3.1 to (see Proposition 3.4). In Subsection 3.2 we prove several combinatorial properties of super-injective simplicial maps . Theorem 3.1 is proved in Subsection 3.3.

3.1 Injective homomorphisms and super-injective simplicial maps of

We define the rank of an orientable surface to be . It is well-known that is the maximal cardinality of a simplex of (). By Corollary 2.9 the same is true for non-orientable surfaces.

Lemma 3.2

Let .

  • We have if and only if there is no simplex in of cardinality containing .

  • We have if and only if there exists a simplex in of cardinality containing and there exists a class , different from , such that every simplex in of cardinality containing also contains .

  • We have if and only if there are two simplices and in of cardinality such that .

{proof}

Suppose that is non-separating. It is easily seen that . Take two disjoint simplices and in of cardinality and set and . Then and .

Suppose that is separating. Let and be the connected components of . It is easily seen that if and otherwise.

Suppose that . Let be a simplex of containing . Set and . We have , hence

Suppose that . Then one of the connected components of , say , is a one-holed Klein bottle. By Lemma 2.10 (2), contains a unique element, . Let be a simplex of of cardinality . Set . Then contains and . Let be a simplex of of cardinality containing . Set and . Since and , we have , hence , and therefore .

Suppose that . It is easily seen that there exist two disjoint simplices and in of cardinality . Similarly, one can find two disjoint simplices and in of cardinality . Set and . Then and are two simplices of of cardinality such that .

For we set . Note that, by Lemma 2.4 and Lemma 2.5, we have for all . We turn now to prove an algebraic characterization of the Dehn twists along the elements of , up to roots and powers. This result and its proof are independent from the characterizations given in Atalan [1, 2] and Atalan–Szepietowski [3], and they are interesting by themselves.

Proposition 3.3
  • Let be an element of infinite order in . If there exist two abelian subgroups and in of rank such that , then there exists such that .

  • Let be a finite index subgroup of , let , and let such that . Then there exist two abelian subgroups and in of rank such that .

{proof}

We take an element of infinite order such that there exist two abelian subgroups and of rank satisfying . First we prove that . Suppose that . This means that all the elements of are one-sided. Let be the reduction homomorphism along . Then is connected and, by Proposition 2.1, is injective. On the other hand the set is an adequate reduction system for all , hence is either periodic or pseudo-Anosov for all . By Lemma 2.11 it follows that . This contradicts the hypothesis which implies .

Now we show that . Suppose that . Then all the elements of are one-sided, is connected, and the reduction homomorphism is injective. Since is an adequate reduction system for , the element is either periodic or pseudo-Anosov. We know by the above that is non-empty, hence we can choose a class . We have , since it is two-sided, hence lies in and it is a reduction class for . So, is periodic, hence is of finite order: contradiction.

Now we show that there exists such that . By the above, we can choose a class . By Proposition 2.7 there exists such that , hence . Similarly there exists such that . So, . Since , it follows that there exists such that .

Since , by Proposition 2.7 there exists a simplex of cardinality containing . By Lemma 3.2 it follows that . Suppose that . Then has two connected components, one of which, , is a one-holed Klein bottle. By Lemma 2.10, is a singleton . If was not an element of , then would be a connected component of and this would contradict Proposition 2.7 (3). Hence . Moreover, again by Proposition 2.7, there exists such that . Similarly, there exists such that . So, . On the other hand, we know that there exist and such that . Then the elements generate a free abelian group of rank lying in : contradiction. So, .

Now we consider a finite index subgroup of , a class , and an integer such that . By Lemma 3.2 there exist two simplices and in of cardinality such that . We assume that . Since is of finite index in , there exists such that for all . Let (resp. ) be the subgroup of generated by (resp. ). Then and are free abelian subgroups of of rank . It remains to show that .

Let . Since , there exist such that . Similarly, there exist such that . By Lemma 2.5 we have . Since , we have for all , hence .

Proposition 3.4

Let be a finite index subgroup of and let be an injective homomorphism. Then there exits a super-injective simplicial map satisfying the following. Let . There exist such that and .

{proof}

Let . Choose such that . By Proposition 3.3 there exist two abelian subgroups of of rank such that . The subgroups and are abelian groups of rank and , hence, by Proposition 3.3, there exists such that . There exist such that