The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Let $N$ be a connected nonorientable surface with or without boundary and punctures, and $j\colon S\rightarrow N$ be the orientation double covering. It has previously been proved that the orientation double covering $j$ induces an embedding $\iota\colon\mathrm{Mod}(N)$ $\hookrightarrow$ $\mathrm{Mod}(S)$ with one exception. In this paper, we prove that this injective homomorphism $\iota$ is a quasi-isometric embedding. The proof is based on the semihyperbolicity of $\mathrm{Mod}(S)$, which has already been established. We also prove that the embedding $\mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F)$ induced by an inclusion of a pair of possibly nonorientable surfaces $F' \subset F$ is a quasi-isometric embedding.


Introduction
Let S = S b g,p be the compact connected orientable surface of genus g with b boundary components and p punctures, and N = N b g,p be the compact connected nonorientable surface of genus g with b boundary components and p punctures.In the case where b = 0 or p = 0, we drop the suffix that denotes 0, excepting g, from S b g,p and N b g,p .For example, N 0 g,0 is simply denoted as N g .If we are not interested in whether a given surface is orientable or not, we denote the surface by F .The mapping class group Mod(F ) of F is the group of isotopy classes of homeomorphisms on 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 This function is independent of the choice of word metrics d H and d G up to Lipschitz equivalence.In addition, there exists a constant K such that δ G H (n) ≤ Kn if and only if the inclusion H ⊂ G is a quasi-isometric embedding.The subgroup H is said to be undistorted in G if this condition is satisfied; otherwise, we say that H is distorted.The distortions of various subgroups in the mapping class groups of orientable surfaces have been extensively investigated.For example, the mapping class groups of subsurfaces are undistorted according to Masur-Minsky [16,Theorem 6.12] and Hamenstädt [10,Proposition 4.1].Farb-Lubotzky-Minsky [7] proved that groups generated by Dehn twists about disjoint curves are undistorted.Moreover, Rafi-Schleimer [19] proved that an orbifold covering map of orientable surfaces induces a quasi-isometric embedding between the mapping class groups.For examples of distorted subgroups of mapping class groups, see Broaddus-Farb-Putman [4], Cohen [5], and Kuno-Omori [13], where it is proved that the Torelli group I b g is distorted in Mod(S b g ).Moreover, it has been proved by Hamenstädt-Hensel [12] that the handlebody group is exponentially distorted in the mapping class group of the boundary surface.
Birman-Chillingworth [2, Theorem 1] (for closed surfaces), Szepietowski [22, Lemma 3] (for surfaces with boundaries), and Gonçalves-Guaschi-Maldonado [8, Theorem 1.1] (for surfaces with punctures) proved that the mapping class group of a nonorientable surface N b g,p is a subgroup of the mapping class group of the orientation double cover S 2b g−1,2p (we will describe the induced injective homomorphism in Section 3).In this paper, we prove the following theorem using the semihyperbolicity of the mapping class group of orientable surfaces, independently established 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, 0, 0), the mapping class group Mod(N b g,p ) is undistorted in the mapping class group Mod(S 2b g−1,2p ).We note that Mod(N 2 ) cannot be embedded in Mod(S 1 ) (see Remark 3.3).The remainder of 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 purely combinatorial group-theoretic terms.We then prepare Lemma 2.8 on the mapping class groups of orientable surfaces, which is used to prove Theorem 1.1.Section 3 is devoted to proving Theorem 1.1.In this section, we also obtain the result that the mapping class groups of nonorientable surfaces are semihyperbolic.Finally, in the appendix, we show that the injective homomorphism between the mapping class groups, which comes from the inclusion of surfaces, is a quasi-isometric embedding (Proposition 4.1).
Acknowledgements: The authors wish to express their great appreciation to B lażej Szepietowski for encouraging the second author to decide whether orientation double coverings induce quasi-isometric embeddings when she visited his office in 2017.Moreover, he pointed out a few mistakes in the proofs of the main results and introduced the authors to Lemma 3.1.His suggestions have considerably improved the proofs and the main results of this paper.The authors express their gratitude to Harry Petyt for variable comments on the semihyperbolicity of extended mapping class groups.The authors are also deeply grateful to Martin Bridson and Saul Schleimer for answering their questions, and to Makoto Sakuma for his comments on an early version of this paper.The first author was supported by JSPS KAKENHI through grant number 20J1431, and the second author was supported by JST, ACT-X, through grant number JPMJAX200D.

Preliminaries
In this section, we show that the centralizer of every element in the (extended) mapping class group of an orientable surface is quasi-isometrically embedded in the (extended) mapping class group.Following Alonso-Bridson [1], we recall the definition of "semihyperbolicity" for finitely generated 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 −1 = {x −1 |x ∈ X} for a set of formal inverses of X. Set A := X ⊔ X −1 .We consider the free monoid A * which consists of all finite words on A. The free group over A is naturally contained in A * , and the natural projection µ : A * → G is well-defined.We denote the length of a word w ∈ A * by ℓ(w).For g ∈ G, we write g (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.An edge of Cay(G, X) corresponds to µ(x) for some x ∈ X, and it is oriented from g to gµ(x) for each g ∈ G.We consider Cay(G, X) as a metric space by assigning a length of 1 to each edge.For g, h ∈ G, the distance in Cay(G, . By this definition of the distance, the group G acts by isometry on Cay(G, X) via the action (g, x) → gx.Let P(Cay(G, X)) be the set of all discrete paths which connect a pair of vertices in Cay(G, X), and let e : P(Cay(G, X)) → Cay(G, X) × Cay(G, X) be the map which sends a discrete path p of length N to (p(0), p(N )).
Definition 2.1.A bicombing σ : Cay(G, X) × Cay(G, X) → P(Cay(G, X)) is a set-theoretic section for e. Namely, σ satisfies e • σ = id.A bicombing σ is said to be quasi-geodesic if there exist λ, ε ≥ 0 such that the image of σ consists only of (λ, ε)-quasi-geodesics.Additionally, a bicombing σ is said to be bounded if there exist k 1 ≥ 1, k 2 ≥ 0 such that, for all x, y, x ′ , y ′ ∈ Cay(G, X) and t ∈ N, We say that a finitely generated group G is weakly semihyperbolic if Cay(G, X) admits a bounded quasi-geodesic bicombing.For elements g, h ∈ G, the image σ(g, h) is called a combing line from g to h.In addition, the combing line σ(1, g) is denoted simply by σ(g).
Alonso-Bridson proved [1] that being a weakly semihyperbolic group is a quasiisometric invariant for finitely generated groups.Therefore, the definition of weak semihyperbolicity for finitely generated groups is independent of the choice of a generating set.Definition 2.2.A bicombing σ for a finitely generated group G is said to be equivariant if σ satisfies the following: for all g ∈ G and x, y ∈ Cay(G, X), g • σ(x, y) = σ(gx, gy).
Here, g • σ(x, y) is the image of the discrete path σ(x, y) under the left action of G on Cay(G, X) and gx, gy are multiplications on the group G.A quasi-geodesic, bounded, and equivariant bicombing for a finitely generated group is called a semihyperbolic structure.A finitely generated group is said to be semihyperbolic if it admits a semihyperbolic structure.
Though the authors do not know whether semihyperbolicity for finitely generated groups is a quasi-isometric invariant, the definition of semihyperbolicity is known to be independent of the choice of a generating set [1, Corollary 4.2].Definition 2.3.Let G be a group with a bicombing σ.Then, a subgroup H ≤ G is said to be σ-quasi-convex if there exists a constant k ≥ 0 such that d(σ(h)(t), H) ≤ k for all h ∈ H and t ∈ N, where σ(h) is the combing line from 1 to h.
An important example of a σ-quasi-convex subgroup is a finite-index subgroup.A finite-index subgroup is σ-quasi-convex because it is dense in the entire group.This fact will be frequently used throughout this paper.
In a semihyperbolic group, every σ-quasi-convex subgroup is quasi-isometrically embedded and inherits the semihyperbolicity.
Lemma 2.4.([1, Lemma 7.2 and Theorem 7.3]) Let G be a finitely generated group with a bicombing σ and let H be a σquasi-convex subgroup of G.Then, the following holds.
(1) If σ is quasi-geodesic, then H is finitely generated and quasi-isometrically embedded in G.
(2) If σ is a semihyperbolic structure for G, then H is semihyperbolic.
In addition, according to [1, Proposition 7.5], centralizers are quasi-convex with respect to a semihyperbolic structure for the entire group.
Lemma 2.5.(Short) Let G be a finitely generated group with a (possibly not quasigeodesic but) bounded equivariant bicombing σ.Then, the centralizer of every element in G is σ-quasi-convex.
Proof of Lemma 2.5.Let a be an element of G.We denote the centralizer of a in G by Z(a).Pick an element g of Z(a).We wish to find a constant k ≥ 0 such that σ(g) is uniformly k-close to Z(a).Let T g be the length of the combing line σ(g).By the equivariance of σ, for each t ≤ T g , there exists an element where k 1 and k 2 are the constants for a bounded bicombing σ.As σ(g)(t)γ t σ(g)(t) −1 = a, there exists a shortest word ψ t such that ψ t γ t ψ −1 t = a.Set A key ingredient of our proof for Theorem 1.1 is that the (extended) mapping class groups of orientable hyperbolic surfaces are semihyperbolic.Remark 2.7.We can see the semihyperbolicity of the mapping class groups Mod(S) of the orientable surface S with non-negative Euler characteristics as follows.There are seven orientable surfaces with non-negative Euler characteristics, S 0 , S 0,1 , S 1 0 , S 0,2 , S 2 0 , S 1 0,1 , and S 1 .Only three of these surfaces have non-trivial mapping class groups, namely, Mod(S 0,2 ) ∼ = Z 2 , Mod(S 2 0 ) ∼ = Z, and Mod(S 1,0 ) ∼ = SL(2, Z).All these groups are Gromov hyperbolic, and so they are semihyperbolic.In these cases, Mod ± (S) is also Gromov hyperbolic, and thus semihyperbolic.
In conclusion, we have the following lemma.Lemma 2.8.Let S be an orientable surface of finite type.The centralizer of every element in the (extended) mapping class group of S is quasi-isometrically embedded and semihyperbolic.

Nonorientable surface mapping class groups are undistorted
In this section, we prove Theorem 1.1.We first explain the orientation double covering of a nonorientable surface.We represent S 2b g−1,2p (g ≥ 1) in the three-dimensional Euclidean space R 3 in such a way that it is invariant under the composition of the three reflections about the xy, yz, and zx planes, as illustrated in Figure 1.Then, we define an involution J :  Proof of Theorem 1.1.As Mod(N 1 ) is trivial, it is clear that Mod(N 1 ) is quasiisometrically embedded in Mod(S 0 ).Thus, we may assume that (g, p, b) is neither (1, 0, 0) nor (2, 0, 0).Set S = S 2b g−1,2p .Because the homeomorphism J defined above is orientation-reversing, the isotopy class [J] is contained in the extended mapping class group Mod ± (S).Let N = N b g,p be the quotient space S/∼ given by the projection j.By Lemma 3.1, Mod(N ) is a subgroup of the centralizer Z([J]) in Mod ± (S).Moreover, as every orientation-preserving homeomorphism of S which commutes with J preserves the fibres of all points of N , all orientation preserving elements in Z([J]) come from Mod(N ), and therefore the index of Mod(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 ambient group.By the definition of f , we have that f (ϕ) = ι(ϕ) for any ϕ ∈ Mod(N ).Note that Mod(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 element, then h ′ (ϕ) = ϕ ∈ Mod(S).If ϕ ∈ Mod ± (S) is an orientation-reversing element, then there exists some ψ ∈ Mod(S) such that ϕ = [J]ψ (i.e.ψ = [J]ϕ), and so we set h ′ (ϕ) = ψ.The composition h ′ •h then gives the identity map on Mod(S).By Lemma 2.8, we see that Z([J]) is quasi-isometrically embedded in Mod ± (S).Let g : Z([J]) ֒→ Mod ± (S) be the quasi-isometric embedding induced by the inclusion map.Then, for any ϕ ∈ Mod(N ), we have (g Consider the composition of h ′ , g, and f : We then have that (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-isometric embeddings, so we are done.
As shown in the proof of Theorem1.1,Mod(N ) is a finite-index subgroup of Z([J]) in the semihyperbolic group Mod ± (S).Hence, by Lemmas 2.4 and 2.5, we have the following corollary.Corollary 3.2.Let N be a nonorientable surface of genus g ≥ 1 with b ≥ 0 boundary components and p ≥ 0 punctures.Then, the mapping class group Mod(N ) is semihyperbolic.
Proof of Corollary 3.2.If N = N 1 , N 2 , the semihyperbolicity of Mod(N ) comes from the fact that it is a finite-index subgroup of the centralizer of [J] in Mod ± (S), and so we only have to prove the assertion for N = N 1 , N 2 .The mapping class groups satisfy Mod(N 1 ) = 1 and Mod(N 2 ) ∼ = Z 2 ⊕ Z 2 , respectively.They are finite groups, so we are done. 6, a 2 b −3 as follows.Suppose, on the contrary, that there exists an injective homomorphism φ : Mod(N 2 ) ֒→ Mod(S 1 ).Let π : Mod(S 1 ) → ā, b | ā2 , b3 = PSL(2, Z) be the canonical projection whose kernel is generated by a 2 .As Kerπ consists of two elements, without loss of generality, we may assume that π(φ(x)) is non-trivial.By the Kurosh subgroup theorem, there exists an element g ∈ PSL(2, Z) such that π(φ(x)) = gāg −1 .Though φ(x) must be of order 2, the elements in the preimage π −1 (gāg −1 ) have order 4, a contradiction.

Appendix
From the work of Masur-Minsky [16] and Hamenstädt's unpublished paper [10], the injective homomorphism between the mapping class groups of orientable surfaces which is induced by an inclusion of the surfaces is a quasi-isometric embedding.In this appendix, we prove a generalization of this result (Proposition 4.1) by using the semihyperbolicity of the mapping class groups.In the following, we do not consider surfaces of infinite type; thus, we assume that any surface has finite genus and 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 have a homomorphism Mod(F ′ ) → Mod(F ) by extending the homeomorphisms of F ′ to the homeomorphisms of F which are trivial on the outside of F ′ .Paris-Rolfsen [18] and Stukow [20] proved that, under the assumption in Proposition 4.1, this natural homomorphism Mod(F ′ ) → Mod(F ) is injective.Proposition 4.1.Let F be a connected orientable or nonorientable surface and F ′ ⊂ F be an admissible connected subsurface.Suppose that every connected component of F − Int(F ′ ) has a negative Euler characteristic.Then, the injective homomorphism Mod(F ′ ) ֒→ Mod(F ) is a quasi-isometric embedding.Proposition 4.1 can be reduced to the following lemma.Lemma 4.2.Let F be a connected orientable or nonorientable surface and F ′ ⊂ F be an admissible connected subsurface.Suppose that every connected component of F − Int(F ′ ) has a negative Euler characteristic.Then, there exists a finite-index subgroup H of Mod(F ) such that the natural injection Mod(F ′ ) ∩ H ֒→ H is a quasi-isometric embedding.
To prove Lemma 4.2, we prepare the following lemmas.Lemma 4.3.Let F be a connected orientable or nonorientable surface of genus g with b ≥ 1 boundary components and p punctures.We assume that b + p ≥ 4 if F is orientable and g = 0. We also assume that g + b + p ≥ 4 if F is nonorientable.Then, there exists a pair {α 1 , α 2 } of essential simple closed curves satisfying the following properties.
(2) F − (IntN (α 1 ) ∪ IntN (α 2 )) is a disjoint union of some copies of N 1 1 , S 1 0 , S 1 0,1 , and S 2 0 .Proof of Lemma 4.3.Suppose that F is orientable.Then, the curve complex of F has infinite diameter (see Theorem 1.1]).This implies that F has a pair {α 1 , α 2 } of essential simple closed curves satisfying condition (2) in Lemma 4.3.We can reduce the case where F is nonorientable to the case where F is orientable by replacing some of the punctures with crosscaps.Then, the pair of closed curves do not pass through a crosscap and each closed curve is two-sided, thereby satisfying condition (1).
Let F be a surface.A closed curve β on F is called peripheral if β is isotopic to a component of ∂F .A two-sided closed curve α on F is called generic if α bounds neither a disk nor a Möbius strip and is not peripheral.Let T (F ) denote the subgroup of Mod(F ), called the twist subgroup, generated by Dehn twists along two-sided closed curves which are either peripheral or generic on F .Lemma 4.4.We have the following.
(3) T (N 1  2 ) ∼ = Z 2 , and its generators are a Dehn twist along a unique peripheral closed curve and a Dehn twist along a unique generic closed curve on N 1 2 .Proof of Lemma 4.4.According to [17,Propositions 17], Mod(N 1  1,1 ) ∼ = Z and is generated by a "boundary slide" s.As the square of s is isotopic to a Dehn twist along a unique peripheral closed curve on N 1  1,1 , the twist subgroup T (N 1 1,1 ) is generated by a Dehn twist.
To obtain an isomorphism Z 2 → T (N 2 1 ), we use the capping homomorphism Mod(N 2 1 ) → Mod(N 1 1,1 ) induced by gluing N 2 1 with a punctured disk along a boundary component C of N 2 1 .Then, the kernel of the capping homomorphism is generated by a Dehn twist along a closed curve isotopic to C. Additionally, the image of a Dehn twist along a peripheral closed curve on N 2 1 which is not isotopic to C is s 2 .Hence, T (N 2 1,0 ) is freely generated by Dehn twists along those peripheral closed curves.By [17,Propositions 22], we have Mod(N 1  2 ) ∼ = Z ⋊ Z.In addition, the first copy of Z is generated by a Dehn twist along a unique generic closed curve on N 1 2 and the second copy is generated by a "crosscap slide" y.As the square of y is isotopic to a Dehn twist along a peripheral closed curve on N 1 2,0 , T (N 1 2 ) is freely generated by those Dehn twists.
The next lemma asserts that the mapping class group of any "essential" subsurface, excepting a few examples, is virtually isomorphic to a direct factor of a σ-quasi-convex subgroup of the ambient mapping class group.Lemma 4.5.Let F be a connected orientable or nonorientable surface and F ′ ⊂ F be an admissible connected subsurface which is not an annulus.Suppose that Mod(F ′ ) = 1 and that every connected component of F − Int(F ′ ) has a negative Euler characteristic.Then, there exist mapping classes ϕ 1 , . . ., ϕ l ∈ T (F ) such that a finite-index subgroup of ∩ l i=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 the index r in Lemma 4.5 is 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 homeomorphic to a one-holed Klein bottle.
Proof of Lemma 4.5.Let F 1 , . . ., F n be the connected components of F − IntF ′ .We denote the genus of F i , the number of boundary components of F i , and the number of punctures of F i as g(F i ), b(F i ), and p(F i ), respectively.As the Euler characteristic of F i is negative, F i satisfies exactly one of the following conditions: (a) F i is orientable and either g( satisfies condition (a) or (c), we have a pair P i of essential closed curves which fills F i in the sense of Lemma 4.3.We define a set of closed curves A i to be a union of P and the set of closed curves of F i which are parallel to ∂F ′ .In the case where F i satisfies condition (b) or (d), the set A i is defined to be the set of closed curves of F i which are parallel to ∂F ′ .Set ϕ α := [T α ] for each α ∈ A := ∪ n i=1 A i and set .
Note that . As 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 ).As all elements in B i are commutative with [T γC ], we have Mod(F ′ ) where r is the sum of the free abelian rank of B 1 , . . ., B n and is equal to the sum of the number of boundary components of F are not contained in F ′ and the number of connected components of F − IntF ′ which are homeomorphic to N 1 2 .In addition, it clearly holds that To simplify the notation, we denote ∩ α∈A Z T (F ) (ϕ α ) by Z.
We now claim that (Mod(F To see this, consider a subset S of Z realizing all possible reversing patterns on orientations of closed curves in A. If there is no element of Z which reverses an orientation of a closed curve in A, we set S = {1}.As 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 so there exists an element s ∈ S such that sf fixes an orientation of each closed curves in A. In the following, we prove that sf ∈ (Mod(F As sf fixes an orientation of each closed curve in A, sf can be decomposed as a product of mapping classes of the regular neighbourhood N (A) of A and F − IntN (A).By Lemma 4.3, F − IntN (A) is a disjoint union of F ′ , outer surfaces F i satisfying condition (b) or (d), and some copies of S 1 0 , S 1 0,1 , S 2 0 , N 1 1 .Obviously, sf | F ′ is contained in Mod(F ′ ).Additionally, if F i satisfies condition (b) or (d), we have that sf | Fi is contained in Mod(F ′ )B i by Lemma 4.4 and the fact that Mod(F i ) is an abelian group freely generated by Dehn twists along peripheral closed curves if F i satisfies condition (b).Note that the copies of S 2 0 are in one-to-one correspondence with the components of ∪ n i=1 ∂F i .Hence, the restriction of sf to the copies of S 1 0 , S 1 0,1 , S 2 0 , and To see this, we use the fact that sf and sf  We are now ready to prove Lemma 4.2.Recall that the mapping class group of an orientable or nonorientable surface is semihyperbolic, and the intersection of two quasi-convex subgroups is also quasi-convex with respect to a semihyperbolic structure (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 case, Proposition 4.1 is trivial.We next consider the case where F ′ is an annulus.Then, Proposition 4.1 can be obtained by using the semihyperbolicity of Mod(F ), because any finitely generated abelian subgroup is quasi-isometrically embedded in a semihyperbolic group (see Theorem 4.10, Chapter III.Γ]).We now assume that Mod(F ′ ) = 1 and F ′ is not an annulus.By Lemma 4.5, there exist mapping classes ϕ 1 , . . ., ϕ l ∈ T (F ) and a non-negative number r such that T (F ′ ) × Z r is naturally embedded in ∩ l i=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.Because F is either closed or an admissible subsurface of some closed nonorientable surface, 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 semihyperbolic structure of T (F ).As each direct factor is quasi-isometrically embedded in a given direct product, the subgroup T (F ′ ) is quasi-isometrically embedded in ∩ l i=1 Z T (F ) (ϕ i ).We then obtain the result that the inclusion map from T (F ′ ) to T (F ) is a quasi-isometric embedding by the σ-quasi-convexity of ∩ l i=1 Z T (F ) (ϕ i ).
Finally, we remark that for closed surfaces, hyperelliptic mapping class groups are also undistorted subgroups because they are centralizers of the mapping class groups (see Stukow [21] for the definition of hyperelliptic mapping class groups of closed nonorientable surfaces).

Lemma 2 . 6 .
([6, Corollary D],[9, Corollary 3.11]) For any orientable hyperbolic surface S of finite type, the mapping class group Mod(S) and extended mapping class group Mod ± (S) of S are semihyperbolic.Here, Mod ± (S) is the group consisting of the isotopy classes of homeomorphisms on S which preserve ∂S pointwise.

Figure 2 .
Figure 2. The orientation double covering j : S 2b g−1,2p → N b g,p F i satisfies condition (a) or (c), the restriction of sf | N (A) to F i is contained in Mod(F ′ ), because sf | N (A) should be trivial on the regular neighbourhood of the filling pairP i .Therefore, sf | N (A) ∈ Mod(F ′ ), and so sf ∈ Mod(F ′ )B 1 • • • B n .Because sf ∈ T (F ), we have that sf ∈ (Mod(F ′ )B 1 • • • B n ) ∩ T (F ), as desired.