The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover
aa r X i v : . [ m a t h . G T ] J a n THE MAPPING CLASS GROUP OF A NONORIENTABLESURFACE IS QUASI-ISOMETRICALLY EMBEDDED IN THEMAPPING CLASS GROUP OF THE ORIENTATION DOUBLECOVER
TAKUYA KATAYAMA AND ERIKA KUNO
Abstract.
Let N be a connected nonorientable surface with or withoutboundary and punctures, and j : S → N the orientation double cover-ing. Birman-Chillingworth, Szepietowski, and Gon¸calves-Guaschi-Maldonadoproved that the orientation double covering j induces an injective homomor-phism ι : Mod( N ) ֒ → Mod( S ) with one exception. In this paper we prove thatthis injective homomorphism ι is a quasi-isometric embedding as an appli-cation of the semihyperbolicity of Mod( S ), which is established by Durham-Minsky-Sisto and Haettel-Hoda-Petyt. We also prove that the embeddingMod( F ′ ) ֒ → Mod( F ) induced by an inclusion of a pair of possibly nonori-entable surfaces F ′ ⊂ F , well-studied by Paris-Rolfsen and Stukow, is a quasi-isometric embedding. Introduction
Let S = S bg,p be a compact connected orientable surface of genus g with b bound-ary components and p punctures, and N = N bg,p a compact connected nonorientablesurface of genus g with b boundary components and p punctures. In the case where b = 0 or p = 0 we drop the suffices which are equal to 0 excepting g from S bg,p and N bg,p . For example, N g, is denoted by N g simply. If we are not interested inwhether a given surface is orientable or not, we denote the surface by F . The map-ping class group Mod( F ) of F is the group of isotopy classes of homeomorphismson F which are orientation-preserving if F is orientable and preserve ∂F pointwise.Recall that if H ⊂ G is a pair of finitely generated groups with word metrics d H and d G (induced by finite generating sets) respectively, then the distortion of H in G is defined as δ GH ( n ) := max { d H (1 , h ) | h ∈ H with d G (1 , h ) ≤ n } . This function is independent of the choice of word metrics d H and d G up to Lips-chitz equivalence. In addition there exists a constant K such that δ GH ( n ) ≤ Kn ifand only if the inclusion H ⊂ G is a quasi-isometric embedding. The subgroup H iscalled undistorted in G if this condition is satisfied; otherwise H is called distorted .The distortions of various subgroups in the mapping class groups of orientable sur-faces have been investigated. For example, the mapping class groups of subsurfaces Date : January 29, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Mapping class group; symmetric mapping class group; nonorientablesurface; semihyperbolicity; subgroup distortion. are undistorted by Masur-Minsky [16, Theorem 6.12] and Hamenst¨adt [10, Propo-sition 4.1]. Farb-Lubotzky-Minsky [7] proved that groups generated by Dehn twistsabout disjoint curves are undistorted. Moreover Rafi-Schleimer [19] proved that anorbifold covering map of orientable surfaces induces a quasi-isometric embeddingbetween the mapping class groups.For examples of distorted subgroups of mapping class groups, Broaddus-Farb-Putman [4], Cohen [5], and Omori and the second author [13] proved that the Torelligroup I bg is distorted in Mod( S bg ). Moreover the handlebody group is exponentiallydistorted in the corresponding mapping class group by Hamenst¨adt-Hensel [12].Birman-Chillingworth [2, Theorem 1] (for closed surfaces), Szepietowski [22,Lemma 3] (for surfaces with boundaries) and Gon¸calves-Guaschi-Maldonado [8,Theorem 1.1] (for surfaces with punctures) proved that the mapping class groupof a nonorientable surface N bg,p is a subgroup of the mapping class group of theorientation double cover S bg − , p (we will describe the induced injective homomor-phism in Section 3). In this paper we show the following as an application of thesemihyperbolicity of the mapping class group of orientable surfaces, independentlyestablished by Durham-Minsky-Sisto [6, Corollary D] and Haettel-Hoda-Petyt [9,Corollary 3.11] Theorem 1.1.
For all but ( g, p, b ) = (2 , , , the mapping class group Mod( N bg,p ) is undistorted in the mapping class group Mod( S bg − , p ) . We note that Mod( N ) cannot be embedded in Mod( S ) (see Remark 3.3).This paper is organized as follows. In Section 2 we first review the definition of“semihyperbolic” groups, which captures the feature of CAT(0) groups in purelycombinatorial group theoretic terms. We then prepare a lemma (Lemma 2.8) onthe mapping class groups of orientable surfaces to prove Theorem 1.1. Section 3is devoted to prove Theorem 1.1. In this section we also obtain the result thatthe mapping class groups of nonorientable surfaces are “semihyperbolic”. As anapplication of the “semihyperbolicity” of the mapping class groups, we show thatthe injective homomorphism between the mapping class groups, which comes froman inclusion of surfaces, is a quasi-isometric embedding (Proposition 4.1).2. Preliminaries
In this section, we show that the centralizer of every element in the mappingclass group of an orientable surface is quasi-isometrically embedded in the mappingclass group. We first recall the definition of the “semihyperbolicity” for finitelygenerated groups.Throughout this section we assume that G is a finitely generated group and X is a finite generating set of G . We write X − = { x − | x ∈ X } for a set of formalinverses of X . Set A := X ⊔ X − . We consider the free monoid A ∗ which consistsof all finite words on A . The free group over A is naturally contained in A ∗ , andthe natural projection µ : A ∗ → G is well-defined. We denote by ℓ ( w ) the length ofa word w ∈ A ∗ . For g ∈ G we write k g k ( G,X ) = min { ℓ ( w ) | w ∈ A ∗ , µ ( w ) = g } . Let Cay(
G, X ) be the Cayley graph of G with respect to a generating set X .Each vertex of Cay( G, X ) corresponds to an element g ∈ G . We write e for theidentity element of G . An edge of Cay( G, X ) corresponds to µ ( x ) for some x ∈ X ISTORTION BETWEEN MAPPING CLASS GROUPS 3 and it is oriented from g to gµ ( x ) for each g ∈ G . We consider Cay( G, X ) asa metric space by assigning length 1 to each edge. For g, h ∈ G the distance inCay( G, X ) is d ( g, h ) = k g − h k ( G,X ) . By the definition of the distance, the group G acts on the Cayley graph Cay( G, X ) on the left by isometries. We shall identifythe word w ∈ A ∗ with the discrete path w : N → Cay(
G, X )such that w (0) = e , w ( t ) = µ ( w ) for t > ℓ ( w ), and if w = a a · · · a n , then w ( i ) isthe vertex µ ( a a · · · a i ) for i = 1 , , · · · n .Let σ : G → A ∗ be a section for µ , that is, σ satisfies µ ◦ σ = id : G → G . Definition 2.1.
A section σ : G → A ∗ is called a combing if there is a positiveconstant K such that for each g ∈ G and x ∈ X , we have d ( σ ( g )( t ) , σ ( gµ ( x ))( t )) ≤ K .For h ∈ G and w ∈ A ∗ , by h · w we denote the path starting at the vertex h andending at the vertex hµ ( w ), which is the image of w under the left action of G onCay( G, X ). Definition 2.2.
A combing σ is called bicombing if there is a positive constant K such that for each g ∈ G and x ∈ X , we have d ( µ ( x ) · σ ( g )( t ) , σ ( µ ( x ) g )( t )) ≤ K .A section σ is called semihyperbolic if σ is bicombing and the image of σ consistsof quasi-geodesics. If σ is a semihyperbolic section of G , then we call σ a semihy-perbolic structure for G . Besides a group G is called semihyperbolic if G admits asemihyperbolic section σ . This definition is equivarent to [3, Definiton 4.1, ChapterIII.Γ] via [3, Proposition 4.5, Chapter III.Γ]. Definition 2.3.
Let G be a group with semihyperbolic structure σ . Then a sub-group H < G is said to be σ -quasiconvex if there exists a constant k > d ( σ ( h )( t ) , H ) ≤ k for all h ∈ H and t ∈ N .Durham-Minsky-Sisto [6] and Haettel-Hoda-Petyt [9] prove that the mappingclass groups of a orientable hyperbolic surfaces of finite types are semihyperbolic: Lemma 2.4. ([6, Corollary D] , [9, Corollary 3.11]) For any orientable hyperbolicsurfaces S of genus g ≥ with b ≥ boundary components and p ≥ punctures,the mapping class group Mod( S ) of S is semihyperbolic. Remark 2.5.
We can see the semihyperbolicity of the mapping class groupsMod( S ) of the orientable surfaces S with non-negative Euler characteristics asfollows. There are seven orientable surfaces with non-negative Euler characteris-tics, S , S , , S , S , , S , S , , and S . The only three surfaces of them havenon-trivial mapping class groups, namely, Mod( S , ) ∼ = Z , Mod( S ) ∼ = Z , andMod( S , ) ∼ = SL(2 , Z ). All these groups are Gromov hyperbolic, and so semihyper-bolic.As shown in a textbook of Bridson-Haefliger [3], we have the following two results: Proposition 2.6. ([3, Proposition 4.12, Chapter III.Γ])
Let G be a semihyperbolicgroup with a semihyperbolic structure σ , and H a σ -quasiconvex subgroup of G .Then H is finitely generated and quasi-isometrically embedded in G , and semihy-perbolic. T. KATAYAMA AND E. KUNO
Proposition 2.7. ([3, Proposition 4.14, Chapter III.Γ])
Let G be a finitely generatedsemihyperbolic group, and let σ be a semihyperbolic structure for G . Then thecentralizer of every element in G is σ -quasiconvex. In conclusion, we have the following key lemma.
Lemma 2.8.
Let S be an orientable surface of genus g ≥ with b ≥ boundarycomponents and p ≥ punctures. The centralizer of every element in the mappingclass group Mod( S ) of S is quasi-isometrically embedded in Mod( S ) . Nonorientable surface mapping class groups are undistorted
In this section we prove Theorem 1.1. Firstly we explain the orientation doublecovering of a nonorientable surface.We represent S bg − , p ( g ≥
1) in the three-dimensional Euclidean space R insuch a way that it is an invariant under the composition of the three reflectionsabout the xy , yz , zx planes, as illustrated in Figure 1. Figure 1.
We represent S bg − , p in R as the left surface (resp. theright surface) when g − g − b circles are the boundary components and 2 p points are the punc-tures of S bg − , p .Then we define an involution J : S bg − , p → S bg − , p by J ( x, y, z ) = ( − x, − y, − z ).The relation ( x, y, z ) ∼ J ( x, y, z ) generates an equivalence relation on S bg − , p . Theprojection j : S bg − , p → S bg − , p / ∼ is a covering map of order two, and the quotientspace S bg − , p / ∼ is the nonorientable surface N bg,p (see Figure 2). In other words,the double covering space of N bg,p by the projection j is S bg − , p ( g ≥ g ≥ g, b ≥
1) and Gon¸calves-Guaschi-Maldonado [8, Theorem 1.1] (for surfaces withpunctures, where g, p ≥
1) proved that the mapping class group Mod( N bg,p ) isa subgroup of Mod( S bg − , p ), which consists of the elements commuting with theisotopy class of J . Moreover, for a nonorientable surfaces N bg,p with p ≥ , b ≥
1, wealso have an injective homomorphism Mod( N bg,p ) ֒ → Mod( S bg − , p ) by applying thesame argument of Szepietowski [22, Lemma 3] to the result of Gon¸calves-Guaschi-Maldonado [8, Theorem 1.1]. We conclude: Lemma 3.1. ([2, Theorem 1] , [22, Lemma 3] , [8, Theorem 1.1]) For all but ( g, p, b ) = (1 , , , (2 , , , the orientation double covering j induces an injective ISTORTION BETWEEN MAPPING CLASS GROUPS 5 homomorphism ι : Mod( N bg,p ) ֒ → Mod( S bg − , p ) . Moreover, the image of Mod( N bg,p ) by ι consists of the isotopy classes of orientation preserving homeomorphisms of S bg − , p which commutes with J . Figure 2.
The orientation double covering j : S bg − , p → N bg,p when g − g − Proof of Theorem 1.1.
Since Mod( N ) is trivial, it is clear that Mod( N ) is quasi-isometrically embedded in Mod( S ). So we may assume ( g, p, b ) is neither (1 , , , , S = S bg − , p . Since the homeomorphism J defined above isorientation reversing, the isotopy class [ J ] is contained in the extended mappingclass group Mod ± ( S ), the group consists of the isotopy classes of all homeomor-phisms on S which preserve ∂S pointwise. Let N = N bg,p be the quotient space S/ ∼ by the projection j . By Lemma 3.1, Mod( N ) is a subgroup of the centralizer Z ([ J ]) in Mod ± ( S ). Moreover, since every orientation preserving homeomorphismof S which commutes with J preserves the fibers of all points of N , the index ofMod( N ) in Z ([ J ]) is two. Then we have the quasi-isometry f : Mod( N ) ֒ → Z ([ J ])induced by ι , because every finite index subgroup is quasi-isometric to the ambi-ent group. By the definition of f , f ( ϕ ) = ι ( ϕ ) for any ϕ ∈ Mod( N ). Note thatMod( S ) is an index two subgroup of Mod ± ( S ). Hence we have the quasi-isometry h : Mod( S ) → Mod ± ( S ) induced by the inclusion map. We define a quasi-inverse h ′ : Mod ± ( S ) → Mod( S ) as follows. If ϕ ∈ Mod ± ( S ) is an orientation preserving el-ement, then h ′ ( ϕ ) = ϕ ∈ Mod( S ). Besides if ϕ ∈ Mod ± ( S ) is an orientation revers-ing element, then there exists ψ ∈ Mod( S ) such that ϕ = [ J ] ψ (i.e. ψ = [ J ] ϕ ), andso we put h ′ ( ϕ ) = ψ . Then the composition h ′ ◦ h is the identity map on Mod( S ). Weremark that the semihyperbolicity is a quasi-isometric invariant (see [1, Theorem1.1]). Hence Mod ± ( S ) is also semihyperbolic by Lemma 2.4. By Propositions 2.6and 2.7, we see that Z ([ J ]) is quasi-isometrically embedded in Mod ± ( S ). We denote T. KATAYAMA AND E. KUNO by g : Z ([ J ]) ֒ → Mod ± ( S ) the quasi-isometric embedding induced by the inclusionmap. Then for any ϕ ∈ Mod( N ), we have ( g ◦ f )( ϕ ) = ( h ◦ ι )( ϕ ) ∈ Mod ± ( S ).Consider the composition of h ′ , g and f ; h ′ ◦ g ◦ f : Mod( N ) ֒ → Z ([ J ]) ֒ → Mod ± ( S ) → Mod( S ) . Then we have ( h ′ ◦ g ◦ f )( ϕ ) = ι ( ϕ ) for any ϕ ∈ Mod( N ). In other words, ι : Mod( N ) ֒ → Mod( S ) decomposes into a composition of three quasi-isometricembeddings, so we are done. (cid:3) Since Mod ± ( S ) is semihyperbolic and by Propositions 2.6 and 2.7, we have thefollowing corollary: Corollary 3.2.
Let N be a nonorientable surface of genus g ≥ with b ≥ boundary components and p ≥ punctures.Then the mapping class group Mod( N ) of N is semihyperbolic.Proof. If N = N , N , the semihyperbolicity of Mod( N ) comes from the fact thatit is a finite index subgroup of the centralizer of [ J ] in Mod ± ( S ), and so we onlyhave to prove the assertion for N = N , N . The mapping class groups satisfyMod( N ) = 1 and Mod( N ) ∼ = Z ⊕ Z , respectively. They are finite groups, so weare done. (cid:3) Remark 3.3.
We can show that Mod( N ) ∼ = h x, y | x , y , [ x, y ] i never be em-bedded in Mod( S ) = h a, b | a , b , a b − i as follows. Suppose, on the con-trary, that there exists an injective homomorphism φ : Mod( N ) ֒ → Mod( S ). Let π : Mod( S ) → h ¯ a, ¯ b | ¯ a , ¯ b i = PSL(2 , Z ) be the canonical projection whose kernelgenerated by a . Since Ker π consists of two elements, without loss of generality weassume that π ( φ ( x )) is non-trivial. By the Kurosh subgroup theorem, there existsan element g ∈ PSL(2 , Z ) such that π ( φ ( x )) = g ¯ ag − . Though φ ( x ) must be oforder 2, the elements in the preimage π − ( g ¯ ag − ) have order 4, a contradiction.4. Appendix
It is known, by Masur-Minsky’s work [16] and Hamenst¨adt’s unpublished paper[10], that the injective homomorphism between the mapping class groups of ori-entable surfaces which is induced by an inclusion of surfaces is a quasi-isometricembedding. We prove a generalization (Proposition 4.1) of this result by using thesemihyperbolicity of the mapping class groups as a goal of this appendix. In thefollowing we do not consider surfaces of infinite type; so we assume that any surfacehas finite genus, finite numbers of boundary components and punctures.Let F be a connected surface. We say that a subsurface F ′ ⊂ F is admissible if F ′ is a closed subset of F . For an admissible subsurface F ′ ⊂ F , we havea homomorphism Mod( F ′ ) → Mod( F ) by extending homeomorphisms of F ′ tohomeomorphisms of F which are trivial on the outside of F ′ . Paris-Rolfsen [18]and Stukow [20] proved under the assumption in Proposition 4.1 that this naturalhomomorphism Mod( F ′ ) → Mod( F ) is injective. Proposition 4.1.
Let F be a connected orientable or nonorientable surface and F ′ ⊂ F an admissible connected subsurface. Suppose that every connected compo-nent of F − Int( F ′ ) has negative Euler characteristic. Then the injective homomor-phism Mod( F ′ ) ֒ → Mod( F ) is a quasi-isometric embedding. Proposition 4.1 can be deduced to the following lemma.
ISTORTION BETWEEN MAPPING CLASS GROUPS 7
Lemma 4.2.
Let F be a connected orientable or nonorientable surface and F ′ ⊂ F an admissible connected subsurface. Suppose that every connected component of F − Int( F ′ ) has negative Euler characteristic. Then there exists a finite indexsubgroup H of Mod( F ) such that the natural injection Mod( F ′ ) ∩ H ֒ → H is aquasi-isometric embedding. In order to prove Lemma 4.2, we prepare the lemmas below.
Lemma 4.3.
Let F be a connected orientable or nonorientable surface of genus g with b ≥ boundary components and p punctures. We assume that b + p ≥ if F is orientable and g = 0 . We also assume that g + b + p ≥ if F is nonorientable.Then there exists a pair { α , α } of essential simple closed curves satisfying thefollowing properties. (1) If F is nonorientable, then the closed curves α , α are two-sided. (2) F − (Int N ( α ) ∪ Int N ( α )) is a disjoint union of some copies of N , S , S , and S .Proof of Lemma 4.3. Suppose that F is orientable. Then as is well-known thecurve complex of F has infinite diameter (see Masur-Minsky [15, Theorem 1.1]).This implies that F has a pair { α , α } of essential simple closed curves satisfyingthe condition (2) in Lemma 4.3.We can deduce the case where F is nonorientable to the case where F is orientableby replacing some of the punctures with crosscaps. Then the pair of closed curvesdo not pass through a crosscap and each closed curve is two-sided, and therebysatisfying the condition (1). (cid:3) Let F be a surface. A closed curve β on F is called peripheral if β is isotopic toa component of ∂F . A two-sided closed curve α on F is called generic if α boundsneither a disk nor a M¨obius strip and is not peripheral. We denote by T ( F ) thesubgroup of Mod( F ), called the twist subgroup , generated by Dehn twists alongtwo-sided closed curves which are either peripheral or generic on F . Lemma 4.4.
We have the following. (1) T ( N , ) ∼ = Z , and that its generator is a Dehn twist along a unique periph-eral closed curve. (2) T ( N ) ∼ = Z , and that its generators are Dehn twists along peripheral closedcurves. (3) T ( N ) ∼ = Z , and that its generators are Dehn twist along a unique periph-eral closed curve and a Dehn twist along a unique generic closed curve on N .Proof of Lemma 4.4. By [17, Propositions 17], Mod( N , ) ∼ = Z and is generated bya “boundary slide” s . Since the square of s is isotopic to a Dehn twist along aunique peripheral closed curve on N , , T ( N , ) is generated by the Dehn twist.In order to obtain an isomorphism Z → T ( N ), we use a capping homomor-phism Mod( N ) → Mod( N , ) induced by gluing N with a punctured disk alonga boundary component C of N . Then the kernel of the capping homomorphismis generated by a Dehn twist along a closed curve isotopic to C . Besides the imageof a Dehn twist along a peripheral closed curve on N which is not isotopic to C is s . Hence T ( N , ) is freely generated by Dehn twists along those peripheral closedcurves. T. KATAYAMA AND E. KUNO
By [17, Propositions 22], Mod( N ) ∼ = Z ⋊ Z . In addition the first copy of Z is generated by a Dehn twist along a unique generic closed curve on N and thesecond copy of is generated by a “crosscap slide” y . Since the square of y is isotopicto a Dehn twist along a peripheral closed curve on N , , T ( N ) is freely generatedby those Dehn twists. (cid:3) The next lemma asserts that the mapping class group of any “essential” sub-surface, excepting a few examples, is virtually isomorphic to a direct factor of a σ -quasiconvex subgroup of the ambient mapping class group. Lemma 4.5.
Let F be a connected orientable or nonorientable surface and F ′ ⊂ F an admissible connected subsurface which is not an annulus. Suppose that Mod( F ′ ) = 1 and that every connected component of F − Int( F ′ ) has negativeEuler characteristic. Then there exists mapping classes ϕ , . . . , ϕ l ∈ T ( F ) suchthat a finite index subgroup of ∩ li =1 Z T ( F ) ( ϕ i ) is isomorphic to T ( F ′ ) × Z r . Here, Z T ( F ) ( ϕ i ) is the centralizer of ϕ i in T ( F ) and r in Lemma 4.5 is the sumof the number of boundary components of F which are not contained in F ′ andthe number of connected components of F − Int( F ′ ) which are homeomorphic to aone-holed Klein bottle. Proof of Lemma 4.5.
Let F , . . . , F n be the connected components of F − Int F ′ .By g ( F i ), b ( F i ) and p ( F i ) we denote the genus of F i , the number of the bound-ary components of F i and the number of the punctures of F i , respectively. Sincethe Euler characteristic of F i is negative, F i satisfies exactly one of the followingconditions:(a) F i is orientable and either g ( F i ) ≥ b ( F i ) + p ( F i ) ≥ F i is orientable, g ( F i ) = 0 and b ( F i ) + p ( F i ) = 3.(c) F i is nonorientable and g ( F i ) + b ( F i ) + p ( F i ) ≥ F i is nonorientable and g ( F i ) + b ( F i ) + p ( F i ) = 3.If F i satisfies the condition (a) or (c), we have a pair P i of essential closed curveswhich fills F i in the sense of Lemma 4.3. We define a set of closed curves A i to bea union of P and a set of closed curves of F i which are parallel to ∂F ′ . In the casewhere F i satisfies the condition (b) or (d), A i is defined to be the set of the closedcurves of F i which are parallel to ∂F ′ . Put ϕ α := [ T α ], where α ∈ A := ∪ ni =1 A i andput B i := ( h [ T β ] | β ∈ ∂F ∩ ∂F i i (if F i = N ) h [ T γ ] | γ is a two sided generic closed curve i (if F i ∼ = N ) . Note that B i ∼ = Z when F i ∼ = N . Consider a subgroup (Mod( F ′ ) B · · · B n ) ∩ T ( F )of T ( F ). Since F ′ is not an annulus and Mod( F ′ ) = 1, for each component C of ∂F ′ ∩ ∂F i there exists a two-sided essential closed curve γ C in F such that γ C intersects C non-trivially in minimal position and is disjoint from ( ∂F ∪ ∂F ′ ) − { C } (and a unique two-sided generic closed curve on F i if F i ∼ = N ). This fact impliesthat, for all i we have Mod( F ′ ) ∩ B i = 1. Therefore (Mod( F ′ ) B · · · B n ) ∩ T ( F ) ∼ = T ( F ′ ) × Z r , where r is the sum of the free abelian rank of B , . . . , B n and is equalto the sum of the number of boundary components of F which are not contained in F ′ and the number of connected components of F − Int F ′ which are homeomorphicto N . In addition it clearly holds that(Mod( F ′ ) B · · · B n ) ∩ T ( F ) ⊂ ∩ α ∈ A Z T ( F ) ( ϕ α ) . ISTORTION BETWEEN MAPPING CLASS GROUPS 9
To simplify the notation we denote ∩ α ∈ A Z T ( F ) ( ϕ α ) by Z .We now claim that (Mod( F ′ ) B · · · B n ) ∩ T ( F ) is a finite index subgroup of Z .To see this, consider a subset S of Z realizing all possible reversing patterns onorientations of closed curves in A . If there exists no element of Z which reverses anorientation of a closed curve in A , we put S = { } . Since A is finite, we can choose S to be finite. Pick an element f in Z . Then f preserves each closed curve in A , and sothere exists an element s ∈ S such that sf fixes an orientation of each closed curvesin A . In the following we prove that sf ∈ (Mod( F ′ ) B · · · B n ) ∩ T ( F ). Then thisimmediately implies that (Mod( F ′ ) B · · · B n ) ∩ T ( F ) is of finite index in Z . Since sf fixes an orientation of each closed curve in A , sf can be decomposed as a productof mapping classes of the regular neighbourhood N ( A ) of A and F − Int N ( A ). ByLemma 4.3, F − Int N ( A ) is a disjoint union of F ′ , outer surfaces F i satisfyingthe condition (b) or (d) and some copies of S , S , , S , N . Obviously sf | F ′ iscontained in Mod( F ′ ). Besides if F i satisfies the condition (b) or (d), we have that sf | F i is contained in Mod( F ′ ) B i by Lemma 4.4 and the fact that Mod( F i ) is anabelian group freely generated by Dehn twists along peripheral closed curves if F i satisfies the condition (b). Note that the copies of S are in 1-1 correspondence tothe components of ∪ ni =1 ∂F i . Hence the restriction of sf to the copies of S , S , , S and N in F − Int N ( A ) is contained in Mod( N ( ∪ ni =1 ∂F i )) ⊂ Mod( F ′ ) B · · · B n by Alexander’s theorem and Epstein’s theorem [17, Proposition 5]. Therefore wehave sf | F − Int N ( A ) ∈ Mod( F ′ ) B · · · B n . Furthermore, we can verify that sf | N ( A ) is contained in Mod( F ′ ). To see this, we use the fact that sf and sf | F − Int N ( A ) are commutative with all of ϕ α ( α ∈ A ). Since sf | N ( A ) = sf · ( sf | F − Int N ( A ) ) − , sf | N ( A ) is also commutative with all of ϕ α ( α ∈ A ). If F i satisfies the condition (b)or (d), the restriction of sf | N ( A ) to F i is contained in Mod( F ′ ), because A i ⊂ F ′ .If F i satisfies the condition (a) or (c), the restriction of sf | N ( A ) to F i is containedin Mod( F ′ ), because sf | N ( A ) should be trivial on the regular neighbourhood of thefilling pair P i . Therefore sf | N ( A ) ∈ Mod( F ′ ), and so sf ∈ Mod( F ′ ) B · · · B n . Since sf ∈ T ( F ), we have sf ∈ (Mod( F ′ ) B · · · B n ) ∩ T ( F ), as desired. (cid:3) Remark 4.6.
Suppose that p ≥
3. If we put F = S ,p +2 and F ′ = S ,p inLemma 4.5, then we conclude that the braid group B p on p -strands is quasiconvexwith respect to a semihyperbolic structure of Mod( S ,p +2 ) because S = { } .We are now ready to prove Lemma 4.2. Recall that the mapping class groupof an orientable or nonorientable surface is semihyperbolic, and the intersectionof two quasiconvex subgroup is also quasiconvex with respect to a semihyperbolicstructure (cf. Bridson-Haefliger [3, Proposition 4.13, Chapter III.Γ]). Proof of Lemma 4.2.
We first consider the case where Mod( F ′ ) = 1. In this caseProposition 4.1 is trivial. We next consider the case where F ′ is an annulus. ThenProposition 4.1 can be obtained by using the semihyperbolicity of Mod( F ), be-cause any finitely generated abelian subgroup is quasi-isometrically embedded in asemihyperbolic group (see Bridson-Haefliger [3, Theorem 4.10, Chapter III.Γ]).We now assume that Mod( F ′ ) = 1 and F ′ is not an annulus. By Lemma 4.5there exists mapping classes ϕ , . . . , ϕ l ∈ T ( F ) and a non-negative number r suchthat T ( F ′ ) × Z r is naturally embedded in ∩ li =1 Z T ( F ) ( ϕ i ) as a finite index subgroup.Lickorish [14] proved that T ( F ) is a finite index subgroup of Mod( F ) if F is closed.Since F is either closed or an admissible subsurface of some closed nonorientablesurface, this Lickorish’s theorem together with Paris-Rolfsen [18] and Stukow [20] implies that T ( F ) is a finite index subgroup of Mod( F ). Let σ be a semihyperbolicstructure of T ( F ). Since each direct factor is quasi-isometrically embedded ina given direct product, the subgroup T ( F ′ ) is quasi-isometrically embedded in ∩ li =1 Z T ( F ) ( ϕ i ). Then we obtain the result that the inclusion map from T ( F ′ ) to T ( F ) is a quasi-isometric embedding by the σ -quasiconvexity of ∩ li =1 Z T ( F ) ( ϕ i ). (cid:3) Finally, we remark that for closed surfaces, hyperelliptic mapping class groupsare also undistorted subgroups of ambient mapping class groups because they arecentralizers of the mapping class groups (see Stukow [21] for the definition of hy-perelliptic mapping class groups of closed nonorientable surfaces).
Acknowledgements:
The authors wish to greatly thank B la˙zej Szepietowskifor encouraging the second author to decide whether orientation double coveringsinduce quasi-isometric embeddings, when she visited his office in 2017. Moreover, hepointed out some mistakes in the proofs of main results, told the authors Lemma 3.1.His valuable suggestions enable them to improve their proofs and results. Theauthors are also deeply grateful to Martin Bridson and Saul Schleimer for answeringtheir questions, and to Makoto Sakuma for pointing out typos in the first draftand suggesting a lot of improvements. The first author was supported by JSPSKAKENHI, the grant number 20J1431, and the second author was supported byJST, ACT-X, the grant number JPMJAX200D, Japan.
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