The proper geometric dimension of the mapping class group of an orientable surface with punctures

Nestor Colin; Rita Jiménez Rolland; Porfirio L. León Álvarez; Luis Jorge Sánchez Saldaña

doi:10.1515/jgth-2024-0227

Artikel Open Access

The proper geometric dimension of the mapping class group of an orientable surface with punctures

Nestor Colin , Rita Jiménez Rolland , Porfirio L. León Álvarez und Luis Jorge Sánchez Saldaña

Veröffentlicht/Copyright: 27. Mai 2025

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Aus der Zeitschrift Journal of Group Theory Band 28 Heft 6

Abstract

We show that the full mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for pure mapping class groups by Aramayona and Martínez Pérez. As a consequence of our result, we also obtain the proper geometric dimension of full spherical braid groups.

1 Introduction

Let S g be an orientable closed connected surface of genus g ≥ 0 , and for n ≥ 1 , consider a collection { p 1 , … , p n } of distinguished points in S g . The mapping class group Mod g n is the group of isotopy classes of all orientation-preserving homeomorphisms f : S g n → S g n , where S g n : = S g − { p 1 , … , p n } . The group Mod g n permutes the set { p 1 , … , p n } and the kernel of this action is the pure mapping class group PMod g n . We denote the group of isotopy classes of orientation-preserving homeomorphisms of S g by Mod g .

In this paper, we compute the minimal dimension gd ¯ ⁡ ( Mod g n ) , often called the proper geometric dimension, of a classifying space E ¯ ⁢ Mod g n for proper actions of Mod g n . Recall that, for a discrete group 𝐺, the space E ¯ ⁢ G is a contractible 𝐺-CW-complex on which G acts properly, and such that the fixed point set of a subgroup H < G is contractible if 𝐻 is finite, and is empty otherwise. Since Mod g n is virtually torsion-free, its virtual cohomological dimension vcd ⁡ ( Mod g n ) is a lower bound for gd ¯ ⁡ ( Mod g n ) . Our main result is the following.

Theorem 1.1

For any g ≥ 0 , n ≥ 1 , the proper geometric dimension of Mod g n is

gd ¯ ⁡ ( Mod g n ) = vcd ⁡ ( Mod g n ) .

Moreover, there exists a cocompact E ¯ ⁢ Mod g n of dimension equal to vcd ⁡ ( Mod g n ) .

The computation of the proper geometric dimension follows from Theorem 3.8 for g = 0 , Theorem 4.3 for g = 1 , Theorem 5.1 for g = 2 , and Theorem 6.2 for g ≥ 3 . These theorems are the main results of Sections 3, 4, 5, and 6, respectively, which provides a fair overview of the structure of the paper. For 2 ⁢ g + n > 2 , the Teichmüller space T g n is known to be a model for E ¯ ⁢ Mod g n ; see for instance [15, Proposition 2.3] and [21, Section 4.10]. In fact, Ji and Wolpert proved in [15, Theorem 1.2] that the truncated Teichmüller space T g n ⁢ ( ϵ ) is a cocompact classifying space for proper actions for 𝜖 sufficiently small. Hence the moreover part of the main result follows from our computation of gd ¯ ⁡ ( Mod g n ) and a result of Lück [20, Theorem 13.19] (see Theorem 2.1 below).

1.1 Known results and filling a gap in the literature

Harer [12, Section 2] constructed a spine for Teichmüller space, that is a PMod g n -equivariant deformation retraction of T g n provided n > 0 . This spine gives a model for E ¯ ⁢ PMod g n , which is of minimal dimension for the mapping class group Mod g 1 of an orientable surface with exactly one puncture; see [12, Theorem 2.1] and [7, Introduction].

On the other hand, Aramayona and Martínez Pérez proved that the mapping class group of any closed orientable surface Mod g (see [2, Theorem 1.1]) admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. They use their result and the Birman exact sequence to show that this is also the case for the pure mapping class group PMod g n (see [2, Corollary 1.3], and also arXiv:1302.4238v.2 and [7, Corollary 3.5]).

However, [2, Theorem 1.1] does not imply the result for the full mapping class group Mod g n when n ≥ 2 , and [2, Theorem 1.1] has been misquoted in the literature. For instance, the existence of a model of minimal dimension for the full mapping class group is used in the proofs of [17, Theorem 1 & Main Theorem], [24, Proposition 5.3 & Theorem 1.4], [1, Theorem 1.1], and [16, Theorem 1.5]. Our Theorem 1.1 fills this gap, and together with [2, Theorem 1.1], it gives the necessary ingredients to prove the following result.

Corollary 1.2

Corollary 1.2 ([24, Proposition 5.3])

Let 𝑆 be a closed (possibly disconnected) surface with a finite number (possibly zero) of punctures. Then there is a cocompact model for E ¯ ⁢ Mod ⁡ ( S ) of dimension gd ¯ ⁡ ( Mod ⁡ ( S ) ) = vcd ⁡ ( Mod ⁡ ( S ) ) .

1.2 Proper geometric dimension of spherical braid groups

The full 𝑛-th strand spherical braid group B n ⁢ ( S 0 ) is the fundamental group of the 𝑛-th unordered configuration space of the sphere S 0 . It is the trivial group for n = 1 and a cyclic group of order 2 for n = 2 . For all n ≥ 1 , these groups are virtually torsion free and their virtual cohomological dimension is

vcd ⁡ ( B n ⁢ ( S 0 ) ) = max ⁡ { 0 , n − 3 }

(see [10, Theorem 5]). For n ≥ 3 , the mapping class group Mod 0 n is related to B n ⁢ ( S 0 ) by the following central extension (see for example [11, Section 2.4]):

1 → Z / 2 → B n ⁢ ( S 0 ) → Mod ⁡ ( S 0 n ) → 1 .

Note that any cocompact model for E ¯ ⁢ Mod ⁡ ( S 0 n ) is, via the above short exact sequence, a cocompact model for E ¯ ⁢ B n ⁢ ( S 0 ) . It follows from [21, Lemma 5.8] that

gd ¯ ( B n ( S 0 ) ) = gd ¯ ( Mod ( S 0 n ) ,

and Theorem 1.1 implies the following corollary.

Corollary 1.3

For any n ≥ 1 ,

gd ¯ ⁡ ( B n ⁢ ( S 0 ) ) = vcd ⁡ ( B n ⁢ ( S 0 ) ) = max ⁡ { 0 , n − 3 } .

Moreover, there is a cocompact model for E ¯ ⁢ B n ⁢ ( S 0 ) of dimension vcd ⁡ ( B n ⁢ ( S 0 ) ) .

1.3 Overview of the proof and the paper

In order to prove our main result Theorem 1.1, we follow the general strategy of [2], and obtain the proper geometric dimension gd ¯ ⁡ ( Mod g n ) by computing its algebraic counterpart cd ¯ ⁡ ( Mod g n ) . This computation amounts to verifying that, for any finite subgroup 𝐹 of Mod g n , the inequality vcd ⁡ ( N ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod g n ) holds, and using [2, Theorem 3.3] (see Theorem 2.2 below). Here N ⁢ F denotes the normalizer of 𝐹 in Mod g n and λ ⁢ ( F ) is the length of 𝐹. In Section 2.1, we introduce the necessary definitions and details.

An application of Nielsen realization and Maher’s Lemma 2.4 implies that vcd ⁡ ( N ⁢ F ) = vcd ⁡ ( Mod g F n F ) for any finite subgroup 𝐹 of Mod g n . The parameters g F and n F are invariants of the orbifold quotient S g n / F as described in Section 2.2. Most of the present paper deals with either computing or upper-bounding the parameters n F and g F . Once we have computed g F and n F , we can use Harer’s computation of the vcd for mapping class groups (2.1) to obtain vcd ⁡ ( Mod g F n F ) . On the other hand, λ ⁢ ( F ) can be bounded by the number of factors in the prime decomposition of | F | .

For g ≥ 3 , we obtain Theorem 6.2 as a straightforward application of [2, Proposition 4.4], which we explain in Section 6. However, for genus g = 0 , g = 1 , and g = 2 , an independent and careful analysis of the finite subgroups of Mod g n is required. This is done in Sections 3, 4, and 5, respectively.

2 Preliminaries

2.1 Proper geometric and cohomological dimensions

There are several notions of dimension defined for a given group 𝐺. We recall in this section the notions that will be used in this paper.

A model for the classifying space of 𝐺 for proper actions E ¯ ⁢ G is a 𝐺-CW-complex 𝑋 such that the fixed point set X H of a subgroup H < G is contractible if 𝐻 is finite, and is empty otherwise. Such a model always exists and is unique up to proper 𝐺-homotopy. The proper geometric dimension of 𝐺 is defined to be

gd ¯ ⁡ ( G ) = min ⁡ { n ∈ Z ≥ 0 : there exists an ⁢ n ⁢ -dimensional model for ⁢ E ¯ ⁢ G } .

On the other hand, let Fin be the collection of finite subgroups of 𝐺, and consider the restricted orbit category O Fin ⁢ G , which has as objects the homogeneous 𝐺-spaces G / H , H ∈ Fin , and morphisms the 𝐺-maps. An O Fin ⁢ G -module is a contravariant functor from O Fin ⁢ G to the category of abelian groups, and a morphism between two O Fin ⁢ G -modules is a natural transformation of the underlying functors. The category O Fin ⁢ G ⁢ -mod of O Fin ⁢ G -modules is an abelian category with enough projectives; see for example [23, p. 7].

The proper cohomological dimension of 𝐺 is defined to be

cd ¯ ( G ) = min { n ∈ Z ≥ 0 : there exists a projective resolution of the O Fin G -module Z Fin of length n } ,

where Z Fin is the constant O Fin ⁢ G -module given by

Z Fin ⁢ ( G / H ) = Z for all ⁢ H ∈ Fin ,

sending every morphism of O Fin ⁢ G to the identity function.

The cohomological dimension cd ⁡ ( H ) of a group 𝐻 is the length of shortest projective resolution, in the category of 𝐻-modules, for the trivial 𝐻-module ℤ. If 𝐺 is virtually torsion free, then 𝐺 contains a torsion free subgroup 𝐻 of finite index, and the virtual cohomological dimension of 𝐺 is defined to be vcd ⁡ ( G ) = cd ⁡ ( H ) . A well-known theorem of Serre establishes that vcd ⁡ ( G ) is well defined and it does not depend on the choice of the finite index torsion free subgroup 𝐻 of 𝐺. For every virtually torsion free group 𝐺, we have the following inequalities (see [5, Theorem 2]):

vcd ⁡ ( G ) ≤ cd ¯ ⁡ ( G ) ≤ gd ¯ ⁡ ( G ) ≤ max ⁡ { 3 , cd ¯ ⁡ ( G ) } .

Following the general strategy of [2], we will determine the proper geometric dimension gd ¯ ⁡ ( Mod g n ) by computing the algebraic counterpart cd ¯ ⁡ ( Mod g n ) . We recall here the two main properties of cd ¯ ⁡ ( G ) that we will need to obtain Theorem 1.1.

First, the following theorem is a consequence of [20, Theorem 13.19], as explained in [2, Section 3]; see also [5, Theorem 1].

Theorem 2.1

Let 𝐺 be a group with cd ¯ ⁡ ( G ) = d ≥ 3 . Then there is a 𝑑-dimensional E ¯ ⁢ G . Moreover, if 𝐺 has a cocompact model for E ¯ ⁢ G , then it also admits a cocompact E ¯ ⁢ G of dimension 𝑑.

Now, let 𝐹 be a finite subgroup of 𝐺. We denote by Z G ⁢ ( F ) , N G ⁢ ( F ) , and W G ⁢ ( F ) = N G ⁢ ( F ) / F the centralizer, normalizer, and Weyl group of𝐹, respectively. If there is no risk of confusion, we will omit the parenthesis and subindices, i.e. we will use the notation Z ⁢ F , N ⁢ F , and W ⁢ F . The length λ ⁢ ( F ) of a finite group 𝐹 is the largest i ≥ 0 for which there is a sequence

1 = F 0 < F 1 < ⋯ < F i = F .

The second result used in obtaining our Theorem 1.1 is the following theorem by Aramayona and Martínez Pérez that gives a criterion to compute the proper cohomological dimension. It is a consequence of a result of Connolly and Kozniewski [8, Theorem A].

Theorem 2.2

Let 𝐺 be a virtually torsion free group such that, for any finite subgroup F ≤ G , vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( G ) . Then cd ¯ ⁡ ( G ) = vcd ⁡ ( G ) .

Remark 1

For 𝐹 a finite subgroup of 𝐺, the normalizer N ⁢ F and the Weyl group W ⁢ F are commensurable; therefore, vcd ⁡ ( N ⁢ F ) = vcd ⁡ ( W ⁢ F ) . In our analysis below, we will obtain upper bounds of vcd ⁡ ( N ⁢ F ) in order to apply Theorem 2.2.

The following lemma will be useful in our computations below. For a given g ∈ G , we denote (respectively) by Z ⁢ ( g ) , N ⁢ ( g ) and W ⁢ ( g ) the centralizer, normalizer and Weyl group of the cyclic subgroup ⟨ g ⟩ in 𝐺.

Lemma 2.3

Let 𝐺 be a virtually torsion free group and 𝐹 a finite subgroup. Then

λ ⁢ ( F ) ≤ log 2 ⁡ ( | F | ) , and
vcd ⁡ ( W ⁢ F ) ≤ min ⁡ { vcd ⁡ ( Z ⁢ ( g ) ) ∣ g ∈ F } .

Proof

Item (1) is [13, Lemma 3.1]. For item (2), note that Z ⁢ F is commensurable with W ⁢ F ; hence they have the same vcd . The conclusion follows from the fact that

Z ⁢ F = ⋂ g ∈ F Z ⁢ ( g )

and the monotonicity of the vcd . ∎

2.2 Mapping class groups

For any g ≥ 0 and n ≥ 0 , the group Mod g n is virtually torsion free and its virtual cohomological dimension was computed by J. L. Harer in [12], i.e.

(2.1) vcd ⁡ ( Mod g n ) = { 0 if ⁢ g = 0 ⁢ and ⁢ n < 3 , n − 3 if ⁢ g = 0 ⁢ and ⁢ n ≥ 3 , 1 if ⁢ g = 1 ⁢ and ⁢ n = 0 , 4 ⁢ g − 5 if ⁢ g ≥ 2 ⁢ and ⁢ n = 0 , 4 ⁢ g − 4 + n if ⁢ g ≥ 1 ⁢ and ⁢ n ≥ 1 .

Let 𝐹 be a finite subgroup of Mod g n . By Nielsen realization (see [9, Theorem 7.2] and [18]), there is a hyperbolic structure for S g n on which 𝐹 acts by orientation-preserving isometries, when χ ⁢ ( S g n ) < 0 .

By a slight abuse of notation, we denote by S g n / F both the orbifold and the underlying topological surface. In what follows, we will use the following notation:

g F is the genus of S g n / F ,
o F is the number of elliptic (orbifold) points of S g n / F ,
n F is the number o F of elliptic points plus the number of orbits of punctures of this action,
( g F ; p 1 F , … , p o F F ) is often called the signature of 𝐹, where p 1 F , … , p o F F are the orders of the elliptic points of S g n / F .

Note that n F ≤ n / | F | + o F .

The following lemma is due to Maher. Together with Harer’s computation (2.1), it gives us a way to compute the vcd of the Weyl group W ⁢ F of any finite subgroup 𝐹 of Mod g n .

Lemma 2.4

Lemma 2.4 ([22, Proposition 2.3])

Let F ≤ Mod g n be a finite subgroup with g F and n F as before. Then N ⁢ F has finite index in Mod g F n F . In particular,

vcd ⁡ ( W ⁢ F ) = vcd ⁡ ( N ⁢ F ) = vcd ⁡ ( Mod g F n F ) .

3 Genus 0

The main result of this section is Theorem 3.8 which proves the case g = 0 of Theorem 1.1. In the first half of this section, we study the finite subgroups of Mod 0 n and their actions on S 0 n ; our analysis starts from Stukow’s classification of maximal finite subgroups of Mod 0 n given in [25]. In the second half of the section, we apply the results in the first half to prove Theorem 3.8. It is worth saying that we dealt with cases 5 ≤ n ≤ 11 one by one, and all the computations of vcd ⁡ ( N ⁢ F ) and λ ⁢ ( F ) are summarized in Appendix A by means of several tables.

3.1 Finite subgroups of Mod 0 n and their realization

We start by recalling Stukow’s classification of maximal finite subgroups of Mod 0 n .

Theorem 3.1

Theorem 3.1 ([25, Theorem 3])

A finite subgroup 𝐹 of Mod 0 n is a maximal finite subgroup of Mod 0 n if and only if 𝐹 is isomorphic to one of the following:

a cyclic group Z / ( n − 1 ) if n ≠ 4 ,
the dihedral group D 2 ⁢ n of order 2 ⁢ n ,
the dihedral group D 2 ⁢ ( n − 2 ) of order 2 ⁢ ( n − 2 ) if n = 5 or n ≥ 7 ,
the alternating group A 4 if n ≡ 4 or 10 ⁢ ( mod ⁢ 12 ) ,
the symmetric group S 4 if n ≡ 0 , 2, 6, 8, 12, 14, 18 or 20 ⁢ ( mod ⁢ 24 ) ,
the alternating group A 5 if n ≡ 0 , 2, 12, 20, 30, 32, 42 or 50 ⁢ ( mod ⁢ 12 ) .

Stukow also studied the conjugacy classes of finite subgroups of Mod 0 n .

Theorem 3.2

Theorem 3.2 ([25, Theorem 4, Corollary 5])

Let 𝐹 be a finite subgroup of Mod 0 n . Then the set of conjugacy classes of subgroups of Mod 0 n isomorphic to 𝐹 has two elements if

F ≅ Z / 2 with 𝑛 even,
F ≅ D 2 ⁢ m with 2 ⁢ m ∣ n or 2 ⁢ m ∣ n − 2 ,

and one element otherwise. In particular, any two maximal finite subgroups of Mod 0 n are conjugate if and only if they are isomorphic.

Since, for a given finite subgroup 𝐹 of Mod 0 n , the number n F only depends on the conjugacy class of 𝐹, we only have to work with a representative of such conjugacy class. Let us describe representatives of the conjugacy classes of maximal finite subgroups of Mod 0 n of types (1)–(3) following the numbering used in Theorem 3.1. For this, let us denote by N and S the north and south poles of S 0 , respectively. Also, given any group isomorphic to D 2 ⁢ m , recall that there is an index 2 cyclic subgroup of order 𝑚; we call this subgroup the subgroup of rotations and every element which does not belong to the subgroup of rotations is called a reflection (all such elements have order 2).

Up to a self-homeomorphism of S 0 n , the action of Z / ( n − 1 ) can be realized as follows: we place one of the punctures on N and the remaining n − 1 on the equator equidistantly. The generator of Z / ( n − 1 ) acts on S 0 n by a rotation of angle 2 ⁢ π / ( n − 1 ) that fixes N and S .
Up to a self-homeomorphism of S 0 n , the action of D 2 ⁢ n can be realized as follows: we place all the punctures along the equator equidistantly. As in the previous case, we have a natural action of the subgroup of rotations Z / n on S 0 n , and a reflection can be realized as a half turn that interchanges N and S and leaves the set of punctures invariant.
The realization of D 2 ⁢ ( n − 1 ) is completely analogous to the previous case, the only difference being that we place 2 punctures on N and S and the remaining n − 2 on the equator.

The numbering in the following proposition is designed to be compatible with that of Theorem 3.1. We will use the above descriptions of the actions of maximal finite groups in the proof of the following statement.

Proposition 3.3

Let 𝐹 be a finite subgroup of Mod 0 n .

If F ≅ Z / m ≤ Z / ( n − 1 ) , then
n F = 2 + n − 1 m .
If F ≅ Z / m is a subgroup of the rotation subgroup Z / n of D 2 ⁢ n , then
n F = 2 + n m .
If F ≅ D 2 ⁢ m ≤ D 2 ⁢ n with m > 1 and when 𝑛 is odd, then
n F = 1 + n + 3 ⁢ m 2 ⁢ m .
When 𝑛 is even and 2 ⁢ m ∣ n ,
n F = { 1 + n + 2 ⁢ m 2 ⁢ m , 1 + n + 4 ⁢ m 2 ⁢ m ,
corresponding to the conjugacy classes of subgroups isomorphic to D 2 ⁢ m .

When 𝑛 is even and 2 ⁢ m ∤ n ,
n F = 1 + n + 3 ⁢ m 2 ⁢ m .
If F ≅ Z / m is a subgroup of the rotation subgroup Z / ( n − 2 ) of D 2 ⁢ ( n − 2 ) , then
n F = 2 + n − 2 m .
If F ≅ D 2 ⁢ m ≤ D 2 ⁢ ( n − 2 ) with m > 1 and when 𝑛 is odd, then
n F = 1 + n − 2 + 3 ⁢ m 2 ⁢ m .
When 𝑛 is even and 2 ⁢ m ∣ ( n − 2 ) ,
n F = { 1 + n − 2 + 2 ⁢ m 2 ⁢ m , 1 + n − 2 + 4 ⁢ m 2 ⁢ m ,
corresponding to the conjugacy classes of subgroups isomorphic to D 2 ⁢ m .

When 𝑛 is even and 2 ⁢ m ∤ ( n − 2 ) ,
n F = 1 + n − 2 + 3 ⁢ m 2 ⁢ m .

Proof

We proceed by cases.

(1) We can realize the Z / m -action placing a puncture in N and the remaining n − 1 equidistantly on the equator. Hence { N } and { S } are both orbits of elliptic points of the Z / m -action, while the orbits of punctures are exactly ( n − 1 ) / m .

(2.1) The Z / m -action can be realized placing all punctures on the equator equidistantly. The conclusion is completely analogous to case (1).

(2.2) Recall that D 2 ⁢ m is generated by the rotation subgroup Z / m ≤ Z / n and a half turn 𝜏 that interchanges N and S . In this case, we have the orbit { N , S } . On the other hand, note that the D 2 ⁢ n -action is free away from the poles and the equator. Hence any elliptic point other than N and S is on the equator. Let 𝑋 be the set of all punctures and all the elliptic points on the equator. Note that, for elements of D 2 ⁢ m , the identity element fixes all points of 𝑋, the nontrivial rotations fix no points in 𝑋, and the 𝑚 reflections each fix two points in 𝑋; hence, by the Burnside lemma, the number of orbits of 𝑋 is | X | + 2 ⁢ m 2 ⁢ m . Therefore, if 𝑒 is the number of elliptic points in 𝑋, we have

n F = 1 + n + e + 2 ⁢ m 2 ⁢ m .

Assume 𝑛 is odd; then the axis points of 𝜏 consist exactly of one puncture 𝑝 and its antipode − p which is an elliptic point. Therefore, there are 𝑚 elliptic points, that is, e = m .

Assume 𝑛 is even. Then the axes of reflections in D 2 ⁢ m are obtained by Z / m -rotating the axis of 𝜏 and taking the bisectors of any two consecutive rotations of the axis of 𝜏.

If 2 ⁢ m divides 𝑛, n / m is even. Let 𝑝 be a fixed puncture and let 𝑡 be a fixed generator of Z / m . Then, between 𝑝 and t ⁢ p , there are n / m − 1 punctures, that is, an odd number of punctures. If 𝜏 fixes two punctures, then each bisector also fixes two punctures. Therefore, there are no elliptic points and e = 0 . If 𝜏 fixes two elliptic points, then each bisector also fixes two elliptic points. Therefore, e = 2 ⁢ m .

If 2 ⁢ m does not divide 𝑛, then n / m is odd and 𝑚 is even. Let 𝑝 be a fixed puncture and let 𝑡 be a fixed generator of Z / m . Then, between 𝑝 and t ⁢ p , there are n / m − 1 punctures, that is, an even number of punctures. Hence, whenever 𝜏 fixes two punctures, each bisector fixes two elliptic points, and whenever 𝜏 fixes two elliptic points, each bisector fixes two punctures. In both scenarios, we conclude e = m .

Cases (3.1) and (3.2) are analogous to cases (2.1) and (2.2), respectively, and are left as an exercise to the reader. ∎

3.2 Proof of the main theorem for genus 0

We compute in this subsection, Theorem 3.8 below, the geometric dimension for Mod 0 n .

Lemma 3.4

If 𝐹 is a finite subgroup of Mod 0 n , then vcd ⁡ ( N ⁢ F ) = max ⁡ { n F − 3 , 0 } .

Proof

By Nielsen realization, 𝐹 acts on S 0 n by orientation-preserving homeomorphisms. As a straightforward application of the Riemann–Hurwitz theorem, we have that S 0 n / F is homeomorphic to a sphere, that is, g F = 0 . Now the result is a consequence of Lemma 2.4 and Harer’s computation of the virtual cohomological dimension for mapping class groups given in (2.1). ∎

Proposition 3.5

Let n ≥ 3 and let g ∈ Mod 0 n be an element of finite order r ≥ 2 . Then r ≤ n and

vcd ⁡ ( W ⁢ ( g ) ) ≤ n r − 1 ≤ n 2 − 1 .

Proof

By [25, Theorem 4, Corollary 5], the element 𝑔 is contained in a unique (up to isotopy) maximal finite cyclic subgroup of order 𝑛, n − 1 , or n − 2 , which by Theorem 3.2 and Theorem 3.1, is the rotation subgroup of one of the maximal subgroups Z / ( n − 1 ) , D 2 ⁢ n , or D 2 ⁢ ( n − 2 ) . Hence, up to conjugation, 𝑔 can be realized as a rotation that fixes N and S and the punctures are on the equator and possibly in one or both poles. Therefore, n F ≤ 2 + n / r , and the first inequality in our statement follows from Lemma 3.4. The second inequality is clear. ∎

Proposition 3.6

Let n ≥ 11 and let g ∈ Mod 0 n be an element of order r ≥ 2 . Then

vcd ⁡ ( W ⁢ ( g ) ) + λ ⁢ ( ⟨ g ⟩ ) ≤ vcd ⁡ ( Mod 0 n ) .

Proof

By Proposition 3.5 and Lemma 2.3, we get

vcd ⁡ ( W ⁢ ( g ) ) + λ ⁢ ( ⟨ g ⟩ ) ≤ n r − 1 + log 2 ⁡ ( r ) ≤ n r − 1 + log 2 ⁡ ( n ) ≤ n 2 + log 2 ⁡ ( n ) .

Since n 2 + log 2 ⁡ ( n ) and vcd ⁡ ( Mod 0 n ) = n − 3 are both increasing functions of 𝑛, it is easy to verify that n 2 + log 2 ⁡ ( n ) ≤ n − 3 for n ≥ 11 . ∎

Theorem 3.7

Let n = 5 or n ≥ 7 , and let 𝐹 be a finite subgroup of Mod 0 n . Then

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 0 n ) = n − 3 .

Proof

For n = 5 or 7 ≤ n ≤ 13 , the conclusion follows from the tables in Appendix A, where vcd ⁡ ( W ⁢ F ) is computed using Proposition 3.3 and Lemma 3.4. The rest of the proof deals with the case n ≥ 14 . By Theorem 3.2, up to conjugation, 𝐹 is a subgroup of one of six possibilities.

F ≤ Z / ( n − 1 ) . The result follows directly from Proposition 3.6 for n ≥ 11 .
F ≤ D 2 ⁢ n . If 𝐹 is cyclic, the result follows from Proposition 3.6 for n ≥ 11 . Assume that F ≅ D 2 ⁢ m with m ∣ n . Then, by Lemma 2.3 and Proposition 3.5, we get
vcd ⁡ ( W ⁢ F ) ≤ vcd ⁡ ( Z / m ) ≤ n m − 1 and λ ⁢ ( F ) ≤ log 2 ⁡ ( m ) + 1 .
Thus
vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ n m + log 2 ⁡ ( m ) ≤ n 2 + log 2 ⁡ ( n ) .
Now note that n 2 + log 2 ⁡ ( n ) ≤ n − 3 for n ≥ 14 .
F ≤ D 2 ⁢ ( n − 2 ) . This case is analogous to the previous one.
F ≤ A 4 . We have λ ⁢ ( F ) ≤ λ ⁢ ( A 4 ) = 3 . By Lemma 2.3 and Proposition 3.5, we get
vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ n 2 + 3 .
Note that n 2 + 3 ≤ n − 3 for n ≥ 10 .
F ≤ S 4 . Analogous to case (4).
F ≤ A 5 . Analogous to case (4).∎

Theorem 3.8

Let n ≥ 0 . Then

gd ¯ ⁡ ( Mod 0 n ) = cd ¯ ⁡ ( Mod 0 n ) = vcd ⁡ ( Mod 0 n ) = max ⁡ { n − 3 , 0 } .

Proof

We separate the proof into various cases.

First assume n = 0 , 1 , 2 , 3 . In this case, Mod 0 n is a finite group and the conclusion follows.

Now assume n = 4 . Since vcd ⁡ ( Mod 0 4 ) = 1 , we know that Mod 0 4 is virtually finitely generated free; hence, by a well-known theorem of Stallings, Mod 0 4 acts properly on a tree and this action provides a model for E ¯ ⁢ Mod 0 4 of dimension 1; therefore, gd ¯ ⁡ ( Mod 0 4 ) = cd ¯ ⁡ ( Mod 0 4 ) = 1 .

Now assume n = 6 . In this case, by Birman–Hilden theory, we have a central extension

1 → Z / 2 → Mod 2 → Mod 0 6 → 1 ,

where the kernel is generated by the hyperelliptic involution of S 2 . By [14, Proposition 4.3], there is a model 𝑋 for E ¯ ⁢ Mod 2 of dimension 3. Therefore, X Z / 2 is a model for E ¯ ⁢ Mod 0 6 of dimension at most 3. We conclude that cd ¯ ⁡ ( Mod 0 6 ) = 3 .

Now assume n = 5 or n ≥ 7 . From Theorem 3.7 and Theorem 2.2, we obtain the proper cohomological dimension. By the Eilenberg–Ganea theorem, we have that gd ¯ ⁡ ( Mod 0 n ) = cd ¯ ⁡ ( Mod 0 n ) , except possibly when n = 5 . To rule out that possibility, we consider the following central extension that arises from Birman–Hilden theory:

1 → Z / 2 → Mod 1 2 → Mod 0 5 → 1 ,

where the kernel is generated by the hyperelliptic involution of S 1 2 . It follows from [21, Lemma 5.8] that gd ¯ ⁡ ( Mod 0 5 ) = gd ¯ ⁡ ( Mod 1 2 ) . Hence gd ¯ ⁡ ( Mod 0 5 ) = 2 follows from our computation gd ¯ ⁡ ( Mod 1 2 ) = cd ¯ ⁡ ( Mod 1 2 ) = 2 in the proof of Theorem 4.3. ∎

4 Genus 1

In this section, we prove Theorem 4.3 which gives the case g = 1 of Theorem 1.1. First we consider finite subgroups of Mod 1 n and their actions on S 1 n .

Theorem 4.1

Let n ≥ 0 , and let 𝐹 be a finite subgroup of orientation-preserving diffeomorphisms of S 1 n . Then 𝐹 is isomorphic to ( Z / s × Z / t ) ⋊ Z / m , where 𝑠 and 𝑡 are positive integers such that s ⁢ t ∣ n and m = 1 , 2, 3, 4 or 6. Moreover, the quotient S 1 / F only depends on 𝑚, and we have the following descriptions of these orbifold quotients:

for m = 1 , S 1 / F is a torus with no elliptic points,
for m = 2 , S 1 / F is a sphere with four elliptic points of orders ( 2 , 2 , 2 , 2 ) ,
for m = 3 , S 1 / F is a sphere with three elliptic points of orders ( 3 , 3 , 3 ) ,
for m = 4 , S 1 / F is a sphere with three elliptic points of orders ( 2 , 4 , 4 ) , and
for m = 6 , S 1 / F is a sphere with three elliptic points of orders ( 2 , 3 , 6 ) .

Proof

Let 𝐹 be as in the statement; by capping the punctures, the action of 𝐹 on S 1 n leads to an action of 𝐹 on the torus S 1 . By the uniformization theorem, there is a metric of constant curvature 0 on S 1 such that the action of 𝐹 is by isometries. Next we can lift this action to the universal covering, that is, there is a group F ̃ acting by orientation-preserving euclidean isometries on R 2 that is a central extension of 𝐹 by a rank 2 group of translations, and such that R 2 / F ̃ is diffeomorphic to S 1 / F . Hence F ̃ is a wallpaper group without reflections or glide reflections, and therefore F ̃ is isomorphic to Z 2 ⋊ Z / m for m = 1 , 2, 3, 4, or 6. This implies that there are positive integers 𝑠 and 𝑡 such that

F ≅ ( Z 2 ⋊ Z / m ) / ( s ⁢ Z ⊕ t ⁢ Z ) ≅ ( Z / s × Z / t ) ⋊ Z / m .

Furthermore, note that Z / s × Z / t acts freely on S 1 and preserves the set of punctures in S 1 n , which is a set with 𝑛 elements; as a conclusion, s ⁢ t ∣ n .

For the moreover part of the statement, observe that S 1 / F is diffeomorphic to R 2 / ( Z 2 ⋊ Z / m ) , and the latter quotients are well known; see for instance [26, Theorem 13.3.6]. ∎

Proposition 4.2

Let n ≥ 2 . Then, for every finite subgroup 𝐹 of Mod 1 n , we have

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 1 n ) = n .

Proof

The conclusion is clear when 𝐹 is the trivial subgroup. Let 𝐹 be a nontrivial finite subgroup of Mod 1 n ; then, by the Nielsen realization theorem, we can realize 𝐹 as a finite group of orientation-preserving diffeomorphisms of S 1 n . We proceed by cases following the notation and conclusions in Theorem 4.1.

F ≅ Z / s × Z / t with s ⁢ t ∣ n . In this case, we have that n F = n / s ⁢ t and S 1 n / F is diffeomorphic to S 1 with n / s ⁢ t punctures. Hence, by Lemma 2.3 (1), we get
vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 1 n F ) + λ ⁢ ( F ) ≤ n s ⁢ t + log 2 ⁡ ( n ) ≤ n 2 + log 2 ⁡ ( n ) .
Now note that n / 2 + log 2 ⁡ ( n ) ≤ n for all n ≥ 0 .
F ≅ ( Z / s × Z / t ) ⋊ Z / 2 . In this case, we have that S 1 / F is diffeomorphic to a sphere with 4 elliptic points. Hence we conclude that n F ≤ 4 + n / s ⁢ t ≤ 4 + n / 2 , and vcd ⁡ ( W ⁢ F ) = n F − 3 . Therefore,
vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ n 2 + 1 + log 2 ⁡ ( 2 ⁢ n ) ≤ n 2 + 2 + log 2 ⁡ ( n ) .
Now note that n / 2 + 2 + log 2 ⁡ ( n ) ≤ n for all n ≥ 5 .
F ≅ ( Z / s × Z / t ) ⋊ Z / m with m = 3 , 4 , 6 . In this case, we have that S 1 / F is diffeomorphic to a sphere with 3 elliptic points. Hence we conclude that n F ≤ 4 + n / s ⁢ t ≤ 3 + n / 2 , and vcd ⁡ ( W ⁢ F ) = n F − 3 . Therefore,
vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ n 2 + log 2 ⁡ ( 6 ⁢ n ) ≤ n 2 + 2 + log 2 ⁡ ( n ) .
Now note that n / 2 + 2 + log 2 ⁡ ( n ) ≤ n for all n ≥ 5 .

To finish the proof, we only have to verify the statement for n = 2 , 3 , 4 . From Theorem 4.1, we can describe explicitly the finite subgroups of Mod 1 n and their quotients, and the analysis can be done case by case; the details are left to the reader. ∎

Theorem 4.3

Let n ≥ 0 . Then

gd ¯ ( Mod 1 n ) ) = cd ¯ ( Mod 1 n ) = vcd ( Mod 1 n ) = n .

Proof

For n = 1 , the result follows since Mod 1 1 ≅ SL ⁢ ( 2 , Z ) is virtually free. For n ≥ 2 , Proposition 4.2 and Theorem 2.2 imply cd ¯ ⁡ ( Mod 1 n ) = vcd ⁡ ( Mod 1 n ) .

By the Eilenberg–Ganea theorem, we have that gd ¯ ⁡ ( Mod 1 n ) = cd ¯ ⁡ ( Mod 1 n ) = n , except possibly for n = 2 . Let us rule out this possibility. We have the following short exact sequence:

1 → B 2 ⁢ ( S 1 ) / Z → Mod 1 2 → Mod 1 → 1 ,

where B 2 ⁢ ( S 1 ) is the braid group over the torus on 2 strands, and 𝑍 is its center. On the other hand, B 2 ⁢ ( S 1 ) / Z ≅ Z / 2 ∗ Z / 2 ∗ Z / 2 ; see for instance [3, Proposition 4.4 (i)]. Note that Z / 2 ∗ Z / 2 ∗ Z / 2 is virtually finitely generated free; hence every finite extension 𝐺 of this group admits a proper action on a tree, which provides a 1-dimensional model for E ¯ ⁢ G . The group Mod 1 also admits a 1-dimensional model for E ¯ ⁢ Mod 1 ; hence, by [21, Theorem 5.16], there is a 2-dimensional model for E ¯ ⁢ Mod 1 2 . This establishes that gd ¯ ⁡ ( Mod 1 2 ) = cd ¯ ⁡ ( Mod 1 2 ) = 2 . ∎

5 Genus 2

In order to compute the proper geometric dimension of Mod 2 n , with n ≥ 1 , we use Broughton’s complete classification, up to topological equivalence, of finite group actions on a genus 2 surface [6, Theorem 4.1 & Table 4]; see Appendix B. Note that there are only finitely many conjugacy classes of finite groups that act on S 2 by homeomorphism. Hence, by Nielsen’s realization theorem, given n ≥ 0 , any finite subgroup 𝐹 of Mod 2 n can be realized, up to conjugation, by one of these finite possibilities. This makes the genus 2 case different in nature to the cases of genus 0 and 1 where the isomorphism types of finite subgroups of the corresponding mapping class groups depend strongly on 𝑛.

Theorem 5.1

Let n ≥ 1 . Then, for every nontrivial finite subgroup 𝐹 of Mod 2 n , we have

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 2 n ) = n + 4 .

In particular, cd ¯ ⁡ ( Mod 2 n ) = gd ¯ ⁡ ( Mod 2 n ) = vcd ⁡ ( Mod 2 n ) = n + 4 .

Proof

Since cd ¯ ⁡ ( Mod 2 n ) ≥ vcd ⁡ ( Mod 2 n ) ≥ 5 , we have cd ¯ ⁡ ( Mod 2 n ) = gd ¯ ⁡ ( Mod 2 n ) for every n ≥ 1 .

Let 𝐹 be a nontrivial finite subgroup of Mod 2 n . By Nielsen realization, 𝐹 acts on S 2 n by orientation-preserving homeomorphisms. Recall from Section 2.2 that n F is bounded above by n / | F | + o F .

By Broughton’s classification, the quotient S 2 n / F can only have genus g F = 0 or g F = 1 . When g F = 0 , we have that vcd ⁡ ( W ⁢ F ) = vcd ⁡ ( Mod 0 n F ) = n F − 3 , and when g F = 1 , then vcd ⁡ ( W ⁢ F ) = vcd ⁡ ( Mod 1 n F ) = n F . In Table 11, we recall the classification from [6, Table 4], and we give explicit upper bounds for λ ⁢ ( F ) , n F , and vcd ⁡ ( W ⁢ F ) . From this, we can see that the inequality

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 2 n ) = n + 4

holds for almost all finite subgroups 𝐹 of Mod 2 n and n ≥ 1 . The only exception is the case of F = GL 2 ⁢ ( 4 ) and n = 1 , which does not occur since F = GL 2 ⁢ ( 4 ) cannot be realized as subgroup of Mod 2 1 , indeed since the only puncture will be a fixed point of the action, but as we can read in the signature of this action in Table 11, there are no fixed points. ∎

Remark 2

From Table 11, we can see that there are several finite subgroups 𝐹 of Mod 2 such that

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≰ vcd ⁡ ( Mod 2 ) = 3 .

For instance, vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) = 4 when F = D 2 ⁢ ( 6 ) . Hence we cannot use this strategy to obtain gd ¯ ⁡ ( Mod 2 ) .

6 Genus at least 3

In this section, we promote the results in [2] for Mod g 0 with g ≥ 3 to Mod g n for g ≥ 3 and n ≥ 1 , using a simple argument.

Proposition 6.1

Proposition 6.1 ([2, Proposition 4.4])

For any g ≥ 3 and any finite subgroup 𝐹 of Mod g 0 , we have vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod g 0 ) .

Theorem 6.2

For any g ≥ 3 , n ≥ 1 , and any finite subgroup 𝐹 of Mod g n , we have

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod g n ) .

In particular, gd ¯ ⁡ ( Mod g n ) = cd ¯ ⁡ ( Mod g n ) = vcd ⁡ ( Mod g n ) .

Proof

Let us consider the Birman short exact sequence (see [4, Theorem 4.3])

1 → B n ⁢ ( S g ) → Mod g n ⁡ → φ ⁢ Mod g 0 → 1 ,

where B n ⁢ ( S g ) is the full braid group over S g on 𝑛 strings.

Let 𝐹 be any finite subgroup of Mod g n . Then, by restriction of the above sequence, we obtain the following short exact sequence:

1 → B n ⁢ ( S g ) ∩ N ⁢ F → N ⁢ F ⁢ → φ ⁢ φ ⁢ ( N ⁢ F ) → 1

and note that φ ⁢ ( N ⁢ F ) ≤ N ⁢ φ ⁢ ( F ) . Therefore,

vcd ⁡ ( N ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( φ ⁢ ( N ⁢ F ) ) + vcd ⁡ ( B n ⁢ ( S g ) ∩ N ⁢ F ) + λ ⁢ ( F ) ( by subadditivity of ⁢ vcd ) ≤ vcd ⁡ ( N ⁢ ( φ ⁢ ( F ) ) ) + vcd ⁡ ( B n ⁢ ( S g ) ) + λ ⁢ ( φ ⁢ ( F ) ) ( by monotonicity of ⁢ vcd ) ≤ vcd ⁡ ( Mod g 0 ) + vcd ⁡ ( B n ⁢ ( S g ) ) ( by Proposition 6.1 ) ≤ 4 ⁢ g − 5 + n + 1 = 4 ⁢ g + n − 4 = vcd ⁡ ( Mod g n ) ( by [19, Theorem 1.2] ) .

Now the result follows. The in particular part follows directly from Theorem 2.2. ∎

Funding source: Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México

Award Identifier / Grant number: PAPIIT IA106923

Funding statement: The first author was funded by SECIHTI through the program Estancias Posdoctorales por México. The third author’s work was supported by UNAM Posdoctoral Program (POSDOC). All authors are grateful for the financial support of DGAPA-UNAM grant PAPIIT IA106923.

A λ ⁢ ( F ) and vcd ⁡ ( W ⁢ F ) for finite subgroups of Mod 0 n for 5 ≤ n ≤ 13

In this appendix, we describe n F , vcd ⁡ ( W ⁢ F ) , and λ ⁢ ( F ) for all the conjugacy classes of finite subgroups 𝐹 of Mod 0 n when 5 ≤ n ≤ 13 .

Polyhedral subgroups of Mod 0 n

We keep the numbering from Theorem 3.1 to analyze the polyhedral subgroups of Mod 0 n for n = 6 , 8 , 10 , 12 .

(3) For n = 10 , up to conjugacy, there is a maximal finite subgroup F ≅ A 4 of Mod 0 n , which can be realized as the symmetry group of a tetrahedron. There are three orbits of points with nontrivial stabilizer in 𝐹: the centers of faces, the centers of edges, and the vertices with orbits of length 4, 6, and 4, respectively. Note that, for n = 10 , there must be two orbits of punctures (one of length 4 and another of length 6) and one orbit of elliptic points; hence n F = 3 .

(4) For n = 6 , 8, or 12, up to conjugacy, there is a maximal finite subgroup F ≅ S 4 of Mod 0 n , which can be realized as the symmetry group of a cube (octahedron). There are three orbits of points with nontrivial stabilizer in 𝐹: the centers of faces, the centers of edges, and the vertices with stabilizers of order 4, 2, and 3, respectively. Note that, for n = 6 , we must place the punctures in the centers of the faces (unique orbit of length 6), for n = 8 , we must place the punctures in the vertices of the cube (unique orbit of length 8), and for n = 12 , the punctures can only be placed in the centers of the edges (unique orbit of length 12). Hence, in S 0 n / F , there is only one orbit of punctures and two orbits of elliptic points and n F = 3 .

Now let F ≅ A 4 ≤ S 4 . It can be realized as subgroup of the symmetry group of a tetrahedron embedded into a cube. For n = 6 , there are two orbits of vertices, one orbit of centers of faces (where we have placed the punctures), and the centers of edges are no longer elliptic points; hence n F = 3 . For n = 8 , there are two orbits of vertices (where we have placed the punctures), one orbit of centers of faces, and the centers of edges are no longer elliptic points; hence n F = 3 . Finally, for n = 10 , there are two orbits of vertices, one orbit of centers of faces, and one orbit of the centers of edges (where we have placed the punctures); hence n F = 4 .

(5) For n = 12 , up to conjugacy, there is a maximal finite subgroup F ≅ A 5 of Mod 0 n , which can be realized as the symmetry group of a dodecahedron (icosahedron). There are three orbits of points with nontrivial stabilizer in 𝐹: the centers of faces, the centers of edges, and the vertices. For n = 12 , we must place the punctures in the centers of the faces (unique orbit of length 12). Hence, in S 0 n / F , there is only one orbit of punctures and two orbits of elliptic points and n F = 3 .

We summarize the conclusions in Table 1. Note that

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 0 n ) = n − 3 for ⁢ n = 8 , 10 , 12 .

However, vcd ⁡ ( W ⁢ S 4 ) + λ ⁢ ( S 4 ) = 4 ≰ vcd ⁡ ( Mod 0 6 ) = 3 .

Table 1

Polyhedral subgroups of Mod 0 n for n = 6 , 8 , 10 , 12 .

𝑛	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
6	S 4	3	0	4
	A 4	3	0	3

8	S 4	3	0	4
	A 4	3	0	3

10	A 4	3	0	3

12	S 4	3	0	4
	A 4	4	1	3
	A 5	3	0	4

Cyclic and dihedral subgroups of Mod 0 n

The following tables for 5 ≤ n ≤ 13 come directly from Proposition 3.3. Note that, in all these cases,

vcd ⁡ ( W ⁢ F ) + λ ⁢ ( F ) ≤ vcd ⁡ ( Mod 0 n ) = n − 3 .

Table 2

Cyclic and dihedral subgroups of Mod 0 5 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 4	3	0	2
	Z / 2	4	1	1

(2.1)	Z / 5	3	0	1

(2.2)	D 2 ⁢ ( 5 )	3	0	2

(3.1)	Z / 3	3	0	1

(3.2)	D 2 ⁢ ( 3 )	3	0	2

Table 3

Cyclic and dihedral subgroups of Mod 0 6 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 5	3	0	1

(2.1)	Z / 6	3	0	2
	Z / 3	4	1	1
	Z / 2	5	2	1

(2.2)	D 2 ⁢ ( 6 )	3	0	3
	D 2 ⁢ ( 3 )	3	0	2
		4	1	2
	D 2 ⁢ ( 2 )	4	1	2

(3.1)	Z / 4	3	0	2
	Z / 2	4	1	1

(3.2)	D 2 ⁢ ( 4 )	3	0	3
	D 2 ⁢ ( 2 )	3	0	2
		4	1	2

Table 4

Cyclic and dihedral subgroups of Mod 0 7 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 6	3	0	2
	Z / 3	4	1	1
	Z / 2	5	2	1

(2.1)	Z / 7	3	0	1

(2.2)	D 2 ⁢ ( 7 )	3	0	2

(3.1)	Z / 5	3	0	1

(3.2)	D 2 ⁢ ( 5 )	3	0	2

Table 5

Cyclic and dihedral subgroups of Mod 0 8 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 7	3	0	1

(2.1)	Z / 8	3	0	3
	Z / 4	4	1	2
	Z / 2	6	3	1

(2.2)	D 2 ⁢ ( 8 )	3	0	4
	D 2 ⁢ ( 4 )	3	0	3
		4	1	3
	D 2 ⁢ ( 2 )	4	1	2
		5	2	2

(3.1)	Z / 6	3	0	2
	Z / 3	4	1	1
	Z / 2	5	2	1

(3.2)	D 2 ⁢ ( 6 )	3	0	3
	D 2 ⁢ ( 3 )	3	0	2
		4	1	2
	D 2 ⁢ ( 2 )	4	1	2

Table 6

Cyclic and dihedral subgroups of Mod 0 9 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 8	3	0	3
	Z / 4	4	1	2
	Z / 2	6	3	1

(2.1)	Z / 9	3	0	2
	Z / 3	5	2	1

(2.2)	D 2 ⁢ ( 9 )	3	0	3
	D 2 ⁢ ( 3 )	4	1	2

(3.1)	Z / 7	3	0	1

(3.2)	D 2 ⁢ ( 7 )	3	0	2

Table 7

Cyclic and dihedral subgroups of Mod 0 10 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 9	3	0	2
	Z / 3	5	2	1

(2.1)	Z / 10	3	0	2
	Z / 5	4	1	1
	Z / 2	7	4	1

(2.2)	D 2 ⁢ ( 10 )	3	0	3
	D 2 ⁢ ( 5 )	3	0	2
		4	1	2
	D 2 ⁢ ( 2 )	5	2	2

(3.1)	Z / 8	3	0	3
	Z / 4	4	1	2
	Z / 2	6	3	1

(3.2)	D 2 ⁢ ( 8 )	3	0	4
	D 2 ⁢ ( 4 )	3	0	3
		4	1	3
	D 2 ⁢ ( 2 )	4	0	2
		5	2	2

Table 8

Cyclic and dihedral subgroups of Mod 0 11 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 10	3	0	2
	Z / 5	4	1	1
	Z / 2	7	4	1

(2.1)	Z / 11	3	0	1

(2.2)	D 2 ⁢ ( 11 )	3	0	2

(3.1)	Z / 9	3	0	2
	Z / 3	5	2	1

(3.2)	D 2 ⁢ ( 9 )	3	0	2
	D 2 ⁢ ( 3 )	4	1	1

Table 9

Cyclic and dihedral subgroups of Mod 0 12 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 11	3	0	1

(2.1)	Z / 12	3	0	3
	Z / 6	4	1	2
	Z / 4	5	2	2
	Z / 3	6	3	1
	Z / 2	8	5	1

(2.2)	D 2 ⁢ ( 12 )	3	0	4
	D 2 ⁢ ( 6 )	3	0	3
		4	1	3
	D 2 ⁢ ( 4 )	4	1	3
	D 2 ⁢ ( 3 )	4	1	2
		5	2	2
	D 2 ⁢ ( 2 )	5	2	2
		6	3	2

(3.1)	Z / 10	3	0	2
	Z / 5	4	1	1
	Z / 2	7	4	1

(3.2)	D 2 ⁢ ( 10 )	3	0	3
	D 2 ⁢ ( 5 )	3	0	2
		4	1	2
	D 2 ⁢ ( 2 )	5	2	2

Table 10

Cyclic and dihedral subgroups of Mod 0 13 .

Proposition 3.3 type	𝐹	n F	vcd ⁡ ( W ⁢ F )	λ ⁢ ( F )
(1)	Z / 12	3	0	3
	Z / 6	3	0	3
	Z / 4	5	2	2
	Z / 2	8	5	1

(2.1)	Z / 13	3	0	1

(2.2)	D 2 ⁢ ( 13 )	3	0	2

(3.1)	Z / 11	3	0	1

(3.2)	D 2 ⁢ ( 11 )	3	0	2

B λ ⁢ ( F ) and vcd ⁡ ( W ⁢ F ) for finite subgroups of Mod 2 n

In this appendix, we use Broughton’s classification [6, Theorem 4.1 & Table 4] to give upper bounds for n F , vcd ⁡ ( W ⁢ F ) , and λ ⁢ ( F ) for all the conjugacy classes of nontrivial finite subgroups 𝐹 of Mod 2 n when n ≥ 0 .

Table 11

Finite nontrivial subgroups of Mod 2 n .

𝐹	\| F \|	S 2 / F	n F ≤	vcd ⁡ ( W ⁢ F ) ≤	λ ⁢ ( F ) ≤
Z / 2	2	( S 0 ; 2 , 2 , 2 , 2 , 2 , 2 )	n 2 + 6	n 2 + 3	1
Z / 2	2	( S 1 ; 2 , 2 )	n 2 + 2	n 2 + 2	1
Z / 3	3	( S 0 ; 3 , 3 , 3 , 3 )	n 3 + 4	n 3 + 1	1
Z / 2 × Z / 2	4	( S 0 ; 2 , 2 , 2 , 2 , 2 )	n 4 + 5	n 4 + 2	2
Z / 4	4	( S 0 ; 2 , 2 , 4 , 4 )	n 4 + 4	n 4 + 1	2
Z / 5	5	( S 0 ; 5 , 5 , 5 )	n 5 + 3	n 5	1
Z / 6	6	( S 0 ; 3 , 6 , 6 )	n 6 + 3	n 6	2
Z / 6	6	( S 0 ; 2 , 2 , 3 , 3 )	n 6 + 4	n 6 +1	2
D 2 ⁢ ( 3 )	6	( S 0 ; 2 , 2 , 3 , 3 )	n 6 + 4	n 6 +1	2
Z / 8	8	( S 0 ; 2 , 8 , 8 )	n 8 + 3	n 8	3
D 2 ̃	8	( S 0 ; 4 , 4 , 4 )	n 8 + 3	n 8	2
D 2 ⁢ ( 4 )	8	( S 0 ; 2 , 2 , 2 , 4 )	n 8 + 4	n 8 +1	3
Z / 10	10	( S 0 ; 2 , 5 , 10 )	n 10 + 3	n 10	2
Z / 2 × Z / 6	12	( S 0 ; 2 , 6 , 6 )	n 12 + 3	n 12	3
D 4 , 3 , − 1	12	( S 0 ; 3 , 4 , 4 )	n 12 + 3	n 12	3
D 2 ⁢ ( 6 )	12	( S 0 ; 2 , 2 , 2 , 3 )	n 12 + 4	n 12 + 1	3
D 2 , 8 , 3	16	( S 0 ; 2 , 4 , 8 )	n 16 + 3	n 16	4
Z / 2 ⋊ ( Z / 2 × Z / 2 × Z / 3 )	24	( S 0 ; 2 , 4 , 6 )	n 24 + 3	n 24	4
SL 2 ⁢ ( 3 )	24	( S 0 ; 3 , 3 , 4 )	n 24 + 3	n 24	4
GL 2 ⁢ ( 4 )	48	( S 0 ; 2 , 3 , 8 )	n 48 + 3	n 48	5

Acknowledgements

The authors thank Omar Antolín Camarena and Jesús Nuñez Zimbrón for useful discussions.

Communicated by: Rachel Skipper

References

[1] J. Aramayona, D. Juan-Pineda and A. Trujillo-Negrete, On the virtually cyclic dimension of mapping class groups of punctured spheres, Algebr. Geom. Topol. 18 (2018), no. 4, 2471–2495. 10.2140/agt.2018.18.2471Suche in Google Scholar

[2] J. Aramayona and C. Martínez-Pérez, The proper geometric dimension of the mapping class group, Algebr. Geom. Topol. 14 (2014), no. 1, 217–227. 10.2140/agt.2014.14.217Suche in Google Scholar

[3] P. Bellingeri and E. Godelle, Positive presentations of surface braid groups, J. Knot Theory Ramifications 16 (2007), no. 9, 1219–1233. 10.1142/S0218216507005762Suche in Google Scholar

[4] J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton University, Princeton, 1974. 10.1515/9781400881420Suche in Google Scholar

[5] N. Brady, I. J. Leary and B. E. A. Nucinkis, On algebraic and geometric dimensions for groups with torsion, J. Lond. Math. Soc. (2) 64 (2001), no. 2, 489–500. 10.1112/S002461070100240XSuche in Google Scholar

[6] S. A. Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991), no. 3, 233–270. 10.1016/0022-4049(91)90021-SSuche in Google Scholar

[7] N. Colin, R. Jiménez Rolland, P. L. León Álvarez and L. J. Sánchez Saldaña, On the dimension of Harer’s spine for the decorated Teichmüller space, preprint (2024), https://arxiv.org/abs/2409.04392; to appear in Homology Homotopy Appl. Suche in Google Scholar

[8] F. Connolly and T. Koźniewski, Finiteness properties of classifying spaces of proper Γ-actions, J. Pure Appl. Algebra 41 (1986), no. 1, 17–36. 10.1016/0022-4049(86)90097-6Suche in Google Scholar

[9] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Math. Ser. 49, Princeton University, Princeton, 2012. 10.1515/9781400839049Suche in Google Scholar

[10] D. L. Gonçalves and J. Guaschi, Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of S 2 and R ⁢ P 2 , Pacific J. Math. 287 (2017), no. 1, 71–99. 10.2140/pjm.2017.287.71Suche in Google Scholar

[11] J. Guaschi and D. Juan-Pineda, A survey of surface braid groups and the lower algebraic 𝐾-theory of their group rings, Handbook of Group Actions. Vol. II, Adv. Lect. Math. (ALM) 32, International Press, Somerville (2015), 23–75. Suche in Google Scholar

[12] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. 10.1007/BF01388737Suche in Google Scholar

[13] C. E. Hidber, L. J. Sánchez Saldaña and A. Trujillo-Negrete, On the dimensions of mapping class groups of non-orientable surfaces, Homology Homotopy Appl. 24 (2022), no. 1, 347–372. 10.4310/HHA.2022.v24.n1.a17Suche in Google Scholar

[14] L. Ji, Well-rounded equivariant deformation retracts of Teichmüller spaces, Enseign. Math. 60 (2014), no. 1–2, 109–129. 10.4171/lem/60-1/2-6Suche in Google Scholar

[15] L. Ji and S. A. Wolpert, A cofinite universal space for proper actions for mapping class groups, In the Tradition of Ahlfors–Bers. V, Contemp. Math. 510, American Mathematical Society, Providence (2010), 151–163. 10.1090/conm/510/10023Suche in Google Scholar

[16] R. Jiménez Rolland, P. L. León Álvarez and L. J. Sánchez Saldaña, Commensurators of abelian subgroups and the virtually abelian dimension of mapping class groups, J. Pure Appl. Algebra 228 (2024), no. 6, Article ID 107566. 10.1016/j.jpaa.2023.107566Suche in Google Scholar

[17] D. Juan-Pineda and A. Trujillo-Negrete, On classifying spaces for the family of virtually cyclic subgroups in mapping class groups, Pure Appl. Math. Q. 12 (2016), no. 2, 261–292. 10.4310/PAMQ.2016.v12.n2.a4Suche in Google Scholar

[18] S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. 10.2307/2007076Suche in Google Scholar

[19] D. Lima Gonçalves, J. Guaschi and M. Maldonado, Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces, Confluentes Math. 10 (2018), no. 1, 41–61. 10.5802/cml.45Suche in Google Scholar

[20] W. Lück, Transformation Groups and Algebraic 𝐾-Theory, Lecture Notes in Math. 1408, Springer, Berlin, 1989. 10.1007/BFb0083681Suche in Google Scholar

[21] W. Lück, Survey on classifying spaces for families of subgroups, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math. 248, Birkhäuser, Basel (2005), 269–322. 10.1007/3-7643-7447-0_7Suche in Google Scholar

[22] J. Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), no. 3, 429–468. 10.1215/00127094-2010-216Suche in Google Scholar

[23] G. Mislin and A. Valette, Proper Group Actions and the Baum–Connes Conjecture, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003. 10.1007/978-3-0348-8089-3Suche in Google Scholar

[24] B. Nucinkis and N. Petrosyan, Hierarchically cocompact classifying spaces for mapping class groups of surfaces, Bull. Lond. Math. Soc. 50 (2018), no. 4, 569–582. 10.1112/blms.12166Suche in Google Scholar

[25] M. Stukow, Conjugacy classes of finite subgroups of certain mapping class groups, Türkiye J. Math. 28 (2004), no. 2, 101–110. Suche in Google Scholar

[26] W. P. Thurston, The Geometry and Topology of Three-Manifolds. Vol. IV, American Mathematical Society, Providence, 2022. Suche in Google Scholar

Received: 2024-12-26

Revised: 2025-04-30

Published Online: 2025-05-27

Published in Print: 2025-11-01

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