Asymptotic mapping class groups of closed surfaces punctured along Cantor sets
Javier Aramayona Louis Funar
 


  



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Asymptotic mapping class groups]Asymptotic mapping class groups of closed surfaces punctured along Cantor sets



Universidad Autónoma de Madrid & ICMAT 
 C. U. de Cantoblanco. 28049, Madrid, Spain
aramayona@gmail.com
  
  Institut Fourier, UMR 5582, Laboratoire de Mathématiques 

Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France
louis.funar@univ-grenoble-alpes.fr


The first author was partially funded by grants RYC-2013-13008 and MTM2015-67781. He also acknowledges support from U.S. National Science Foundation grants DMS
1107452, 1107263, 1107367 "RNMS: Geometric Structures and Representation Varieties" (the
GEAR Network). This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 777822.






We introduce subgroups _g< _g of the mapping class group (Σ_g) of a closed surface of genus g ≥ 0 with a Cantor set removed, which are extensions of Thompson's group V by a direct limit of mapping class groups of compact surfaces of genus g. 

We first show that both _g and _g  are finitely presented, and that _g is dense in (Σ_g). 
We  then exploit the relation with Thompson's groups to study properties _g and _g  in analogy with known facts about finite-type mapping class groups. For instance, their  homology coincides with the stable  homology of the mapping class group of genus g, every automorphism is geometric, and every homomorphism from a higher-rank lattice has finite image.

In addition, the same connection with Thompson's groups will also prove that _g and _g are  not linear and do not have Kazhdan's Property (T), which represents a  departure from the current knowledge about finite-type mapping class groups. 















[
    Louis Funar
    December 30, 2023
=====================





§ INTRODUCTION
  
There has been a recent surge of activity around mapping class groups of infinite-type surfaces, namely those whose fundamental group is not finitely generated. These groups share many properties with their finite-type counterparts (e.g. <cit.>), but also show rather different behaviour (e.g. <cit.>). 

Here we will focus our attention on the mapping class group (Σ_g) of the surface Σ_g, namely the closed orientable surface of genus g≥ 0 with a Cantor set C removed. This group is related to the homeomorphism group (C) through the short exact sequence (see Section <ref> for details):

    1 →(Σ_g) →(Σ_g) →(C) → 1,

where (Σ_g) is the pure mapping class group, namely the subgroup of (Σ_g) whose elements fix C pointwise. 

 In this article we study two countable subgroups _g < _g < (Σ_g), whose elements asymptotically preserve a rigid structure on Σ_g. We remark that _0 and _0 were previously introduced in <cit.> under the names  and ^1/2, respectively. We now give a brief description of these groups, referring the reader to Section <ref> for a complete definition.  
 





 
 We first need to introduce the notion of a rigid structure; for this purpose, it will be convenient to start with the genus-zero case.  The reader should keep  Figure <ref> in mind. 
  A rigid structure on Σ_0 is a triple (P,𝒜, Σ_0^+), where:
  
  
  * P is a pants decomposition of Σ_0,
  
  * 𝒜 is a set of pairwise-disjoint properly embedded arcs such that   (the closure of) every connected component of S- P, referred to as a pair of pants,  is intersected by exactly three arcs in 𝒜, and
  
  * Σ_0^+ is a choice of one of the two connected components of Σ_0 - ⋃_a∈𝒜 a.

  
  
 
 
 
 
Let Σ_0^1 be the result of puncturing Σ_0 once. A rigid structure on Σ_0^1 is a rigid structure on the surface (homeomorphic to Σ_0)  obtained by filling in the isolated puncture of Σ_0^1. 
 
 A rigid structure on Σ_g (g ≥ 1) consists of a simple closed curve α which cuts off a once-puncture surface of genus g, together with a rigid structure for the component of Σ_g - α homeomorphic to Σ_0^1. 
 
 

The group _g. Fix, once and for all, a rigid structure on Σ_g. A homeomorphism f:Σ_g →Σ_g is asymptotically rigid if it preserves  the rigid structure outside some compact subsurface Z(f) of Σ_g; see Section <ref> for a complete definition. We define  _g as the subgroup of (Σ_g) whose elements have at least one asymptotically-rigid representative. Obviously, every element of (Σ_g) with compact support belongs to _g, although the converse is not true. Indeed, denote by _c(Σ_g) the subgroup of (Σ_g) generated by compactly-supported elements, which is a direct limit of mapping class groups of compact genus-g subsurfaces of Σ_g.  In Proposition <ref> we will generalize a result of <cit.> to prove that the sequence (<ref>), when restricted to _g, reads:

    1 →_c(Σ_g) →_g → V → 1,
 
where the rightmost non-trivial group is Thompson's group V (see e.g. <cit.>).  This sequence reveals a fascinating connection between mapping class groups and Thompson's groups, and will be a key ingredient in the proofs of most of our results.  



We remark that the notation _c(Σ) is somewhat redundant, for if an element of (Σ) has compact support then it automatically belongs to (Σ). However, we will use the notation _c(Σ) to emphasize the connection with the sequence (<ref>)





The group _g. The group _g is defined in a similar fashion: its elements are those mapping classes which have a representative which preserve (P,𝒜) outside some compact subsurface. Observe that _g < _g, although the inclusion is strict: for instance, a half-twist lies in _g ∖_g. For this reason, the group _g is sometimes referred to as the group of half-twists <cit.>. Using the same reasoning as above, equation (<ref>) restricts to a short exact sequence 

    1 →_c(Σ_g) →_g → V_2[ℤ_2] → 1,

where V_2[ℤ_2] is the Higman-Thompson group V_2[ℤ_2] (see <cit.> and Section <ref>). A surprising result of Bleak-Donoven-Jonušas <cit.> asserts that V and V_2[ℤ_2] are conjugate as subgroups of (C) through an explicit homeomorphism of C (a cellular automaton). 







A large part of the motivation for considering _g comes from the study of smooth mapping class groups. Put a differentiable structure on the closed orientable surface S_g of genus g, and realize C as the the middle-third Cantor set on a smoothly-embedded interval on S_g. Let ^s(S_g, C) denote the smooth mapping class group of the pair (S_g,C), namely the group of isotopy classes of smooth diffeomorphisms of S_g preserving globally the Cantor set C. The following is a recent result of Neretin and the second author <cit.>: 

[<cit.>, Cor. 2]
For every g ≥ 0, we have _g ≅^s(S_g,C).


In particular, ^s(S_g,C) is countable; observe that, in stark contrast, the “topological” mapping class group (Σ_g) is uncountable. 



 §.§ Results
 As we will see, the topological restrictions on the elements of _g and _g impose strong finiteness conditions on the groups. More concretely, we will prove:


For every g≥ 0,   _g and _g are finitely presented. 



We remark that the case g=0 of the above theorem was proved by Kapoudjian and the second author in <cit.>; in fact, it will serve as the base case for the inductive argument behind the proof of Theorem <ref>. 

In spite of the above result, we will prove that _g serves as a good approximation for the mapping class group: 



For every g≥ 0, _g is dense in (Σ_g).



This theorem should be compared with a recent result of Patel-Vlamis <cit.> which asserts that _c(Σ_g) is dense in (Σ_g)







Next, we turn our attention to the study of properties of the groups _g and _g, through the comparison with known/expected/desired properties of mapping class groups of finite-type surfaces.





  §.§.§ Homological stability

Let S_g,n denote the surface of genus g with n boundary components. A celebrated result of Harer <cit.> asserts that, for a fixed genus g≥ 3, the k-th homology group of the mapping class group (S_g,n) is independent of n, provided that k is “sufficiently small” with respect to g (by a result of Boldsen <cit.>, k ≤ 2g/ 3 suffices). For this reason, this homology group is called the k-th stable homology group of the mapping class group of genus g. 
Using a translation of the proof of <cit.> to our setting, we will show: 






For every k ≤ 2g/3,  H_k(_g, ) and H_k(_g, ) are isomorphic to the k-th stable homology group of the mapping class group of genus g. 

 

Powell <cit.> proved that (S_g) is perfect, i.e. has trivial abelianization. As a 
consequence of  the proof of Theorem <ref>, we have:


_g and _g are perfect for every g≥ 3. 











  §.§.§ Isomorphic classification
 
With a few well-understood exceptions,
mapping class groups of finite-type surfaces are isomorphic if and only if the underlying surfaces are homeomorphic. To see this, one may compare the  virtual cohomological dimension of the mapping class group <cit.> with the maximal rank of a free-abelian subgroup <cit.>. In the case of the groups _g and _g both these quantities are infinite.  However, we will prove: 



If 0≤ g<h and 2≤ h, then there are no surjective homomorphisms _h →_g. 



As will become transparent, the same argument will yield that there are no surjective maps _h →_g (resp. _h →_g, and _h →_g). 
As a consequence of Theorem <ref>, _g ≅_h (resp. _g ≅_h) if and only if g = h. In light of this, an obvious question is: 


Are _g and _g isomorphic? 


We stress that, although the results from <cit.> might suggest a  positive answer to the question above, the answer remains unknown for all values of g. In a more general situation, however, if we replace the binary Cantor set C by the set of ends of a regular tree of valence higher than 3 then the corresponding Thompson groups groups V and V[_2] might be non-isomorphic; compare with <cit.>.   




  §.§.§ Rigidity
 A celebrated theorem of Ivanov <cit.> states that the mapping class group of a (sufficiently complicated) finite-type surface is rigid: every automorphism is induced by a surface homeomorphism. This has recently been extended to the infinite-type setting by Patel-Vlamis <cit.> and Bavard-Dowdall-Rafi <cit.>.
Along similar lines, our next result asserts that _g and _g are also rigid. Given a group G and a subgroup H, write (G) for the automorphism group of G, and denote by N_G(H) the normalizer of H in G. We have: 




For every g≥ 0, (_g) ≅ N_(Σ_g)(_g) and (_g) ≅ N_(Σ_g)(_g). 





An immediate consequence of Ivanov's theorem in the finite-type case is that the outer automorphism group of (S_g,n) is finite; in fact, it is trivial for all but finitely many surfaces. However, in Corollary <ref> we will see that this is no longer true for the groups _g and _g. Denote by (G) the outer automorphism group of a group G, that is, the group of conjugacy classes of automorphisms of G. We will prove: 


For every g≥ 0, (_g) and (_g) are  infinite. 




Let P be the pants decomposition underlying the rigid structure on 
Σ_0^1, and consider the element t_∞∈(Σ_0^1) obtained as the product of all (say left) half-twists about the curves of P. 
We further embed Σ_0^1 in Σ_g in such way that the pants decomposition P underlying the rigid structure of Σ_0^1 is sent to the pants decomposition underlying the rigid structure of Σ_g. We will show later (see Lemma <ref>) that the image of t_∞ under the homomorphism induced by the embedding Σ_0^1 →Σ_g produces an infinite-order element 
of N_(Σ_g)(_g) lying in the kernel of the homomorphism (_g)→(V).




  §.§.§ Homomorphisms from lattices
 Building up on work of Ivanov <cit.> and Kaimanovich-Masur <cit.>, Farb-Masur <cit.> proved that any homomorphism from a higher-rank lattice to a finite-type mapping class group has finite image (see also <cit.> for different proofs of this result). Using this result, we will prove:


Let Γ be a lattice in a  semisimple Lie group G of real rank at least two, where G has no compact factors isogenous to SU(1,n) or SO(1,n). 
For every g≥ 0, any homomorphism from Γ to _g or _g has finite image. 





  §.§.§ Kazhdan's Property (T)
 A compactly generated group has Kazhdan's Property (T) if every unitary representation that has almost invariant vectors also has an invariant vector. Since we will not need any further details about Property (T), we simply refer the reader to the book <cit.> for details. 
It is expected <cit.> that mapping class groups of finite-type surfaces do not have Property (T). Using similar arguments to the ones in the proof of Theorem <ref>, we will observe: 


 _g and _g do not have Kazhdan's Property (T) for any g≥ 0. 





  §.§.§ Non-linearity
 A well-known open question asks  whether finite-type mapping class groups are  linear. The only known result in this direction is a theorem of Bigelow-Budney <cit.>, who proved that the mapping class group of the closed surface of genus two is linear, by exploiting its relation with braid groups. 
In sharp contrast, in Proposition <ref> we will see that _g contains an isomorphic copy of Thompson's group F, for all g≥ 0. Combining this with Theorem <ref>, we have: 


_g is not linear for any g≥ 0. In particular, _g is not linear either. 















Plan of the paper.  In Section <ref> we give some basic definitions and set the notation used in the rest of the paper. In Section <ref> we will define the groups _g and _g. In Section <ref> we will explore their relation to  Thompson's group V, which will be a key ingredient in the proof of many of our main results. We then proceed to prove the results mentioned in the introduction. In this direction, Theorem <ref> will be proved in Section <ref>, and Theorem <ref> in Section <ref>.  Section  <ref>. will deal with the proof of Theorem <ref>. In Section <ref> we will prove Theorem <ref>, while Section <ref> is concerned with the proof of Theorem <ref>. In Section <ref> we will establish Theorems <ref> and <ref>. 


 




Acknowledgements. We thank Y. Antolín,  J. Bavard, L. Bowen, J. Hernández, C. Kapoudjian, T. Koberda, C. Martínez-Pérez, H. Parlier, P. Patel, J. Souto, and N. Vlamis for conversations. We are indebted to Juan Souto for suggesting the proof of Lemma <ref> and to the referee for carefully reading our paper and his numerous comments and corrections. Parts of this paper were written while the first author was visiting Yale University, to which he is grateful for its hospitality.


§ PRELIMINARIES

 

In this section we recall some of the basic definitions about surfaces and their mapping class groups. 



 §.§ Curves and surfaces
 Let S be a connected orientable surface, of finite or infinite topological type. If S has punctures, we will regard them either as marked points on S or as topological ends of S, and we will feel free to switch between the two viewpoints without any further mention. We will denote by S_g,n^p the compact surface of genus g≥ 0 with n≥ 0 boundary components and p≥ 0 punctures. Similarly, let Σ_g,n^p the closed surface of genus g≥ 0 with a Cantor set removed, with n≥ 0 boundary components and p≥ 0 isolated punctures. If either n or p is equal to zero we will simply omit it from the notation.  

 By a curve on S we mean the isotopy class of a simple closed curve on S which does not bound a disk, a once-punctured disk, or an annulus whose other boundary curve is a boundary component of S. We say that two curves are disjoint if they have disjoint representatives on S; otherwise we say that they intersect.  Given curves a, b⊂ S, we define their intersection number i(a,b) as the minimal number of points of intersection  between representatives. 
Note that i(a,b) is always finite, even on a surface of infinite topological type, as curves are compact.
 
 A multicurve on S is a set of pairwise disjoint curves on S. We say that a multicurve M is locally finite if any compact subsurface of S intersects only finitely many elements of M.  A pants decomposition of S is a locally-finite multicurve that is maximal with respect to inclusion; as such, its complement on S is a disjoint union of 3-holed spheres, or pairs of pants. 
 
 Finally, an arc on S is a non-trivial isotopy class of properly embedded arcs on S.






























 §.§ Mapping class group
 Let S be a connected orientable surface, possibly of infinite type. The mapping class group (S) is the group of isotopy classes of self-homeomorphisms of S, where homeomorphisms and isotopies are required to fix the boundary of S pointwise. 
We record the following immediate observation for further use: 


For g≥ 0 and n≥ 1,  (S_g,n)<(Σ_g).


In what follows, we will need to make use of the following further subgroups of (S). The pure mapping class group  (S) is the subgroup of (S) whose elements fix every topological end of S. The compactly supported mapping class group _c(Σ_g) is the subgroup of (Σ_g) whose elements are the identity outside a compact subsurface of Σ_g. The following is an easy observation: 


 Let g≥ 0. Consider any family {S_i} of compact subsurfaces of Σ_g whose union equals Σ_g, partially ordered with respect to inclusion. Then 
_c(Σ_g) ≅ ((S_i)).












§ ASYMPTOTIC MAPPING CLASS GROUPS
 
 
In this section we define the groups _g and _g. 
We start by introducing the notion of rigid structure, which appeared originally in <cit.>. 



 §.§ Rigid structures
 As in the introduction, it will be convenient to start with the genus-zero case. 
The reader should keep Figure <ref> in mind. A rigid structure on Σ_0 is a triple (P,A,Σ_0^+), where:


  * P is a pants decomposition of Σ_0, called the  pants decomposition  underlying the rigid structure,

  * A is a set of pairwise-disjoint arcs on Σ_0 such that for every pair of pants Y of P (that is, the closure of a  connected component of Σ_0 - P), there are exactly three elements of A intersecting Y essentially, each  connecting a different pair of boundary curves of Y,

  * Σ_0^+ is one of the two connected components of Σ_0 - ⋃_a ∈ A a, called the visible side of Σ_0. 





Observe that, up to the action of (Σ_0), there is only one rigid structure on Σ_0. 

 It will be useful  to extend the definition of  rigid structure to the surface Σ_0^p obtained from Σ_0 by removing a finite collection of p≥ 1  points.  In this case, we define a rigid structure on  Σ_0^p as a rigid structure on the surface  (homeomorphic to Σ_0) obtained from Σ_0^p by filling every isolated puncture, subject to the condition that every isolated puncture of Σ_0^p is contained in the same connected component of Σ_0 - P, where P is the pants decomposition underlying the rigid structure of Σ_0. 

Finally, consider the case of 1≤ g<∞. A rigid structure on Σ_g  consists of a curve c ⊂Σ_g that cuts off a disk containing every puncture of Σ_g, together with a rigid structure for the planar component of Σ_g - c, namely the one  homeomorphic  to Σ_0^1.  




 §.§ The groups _g and _g
 We now define the groups _g and _g. As mentioned above, we stress that the case g=0 was previously introduced  in <cit.>.

Fix, once and for all, a rigid structure on Σ_g, and write P for   the pants decomposition underlying it. 
We say that a compact subsurface Z ⊂Σ_g is P-suited if 
∂ Z ⊂ P, namely each boundary curve of Z is an element of the pants decomposition P. 





A homeomorphism f:Σ_g →Σ_g is asymptotically rigid if there exists a P–suited genus-g subsurface Z ⊂Σ_g with f(Z) also P–suited,  and such that the restriction homeomorphism f: Σ_g - Z →Σ_g - f(Z) sends: 


  * P ∩ (Σ_g- Z) to  P ∩ (Σ_g- f(Z)),

  * A ∩ (Σ_g- Z) to  A ∩ (Σ_g- f(Z)),

  * The visible side of Σ_g-Z to the visible side of Σ_g-f(Z). 

If we drop the last requirement we say that f is asymptotically quasi-rigid.



Notation. Given an asymptotically rigid (resp. quasi-rigid) homeomorphism f as above, we will refer to the subsurface Z in the definition above as a defining subsurface for f. 



Observe that if f:Σ_g →Σ_g is asymptotically (quasi-) rigid and Z is a defining surface for f, then any P-suited surface containing Z is also a defining surface for f. This observation will be heavily used in the rest of the paper, without further mention. 







We are finally in a position to define the groups we are interested in: 


Let g,p≥ 0. We define _g (resp. _0^p) as the  subgroup of (Σ_g) (resp. (Σ_0^p)) consisting of those elements that have at least one asymptotically rigid representative. 

In turn, the group _g is the subgroup of (Σ_g) consisting of those elements that have at least one asymptotically quasi-rigid representative. 
 

Observe that _c(Σ)⊂_g, by definition. However, the inclusion is proper, as a general element of _g may permute the components of the complement of every defining subsurface.

  





The group of mapping classes of asymptotically quasi-rigid homeomorphisms of the disk punctured along a Cantor set coincides with the braided Thompson group considered by Brin <cit.> and Dehornoy <cit.>. 
The subgroup BV of mapping classes of asymptotically rigid homeomorphisms of the disk punctured along a Cantor set can be realized as a subgroup of _0 (see <cit.>, section 7).  











§ THE RELATION WITH THOMPSON'S GROUPS



































As mentioned in the introduction, the groups _g and _g are strongly related to Thompson's groups. This is a manifestation of a more general phenomenon, which we now explain. 

Observe that any homeomorphism of Σ_g induces a homeomorphism of the space of ends (see, for instance, <cit.>) of Σ_g, which by definition is the Cantor set  C. Thus we have a 
continuous homomorphism 

    ϵ: (Σ_g) →(C),

which is surjective when both homeomorphism groups are endowed with their respective compact-open topologies.





Now, every Cantor set on the plane is tame, meaning that, up to homeomorphism, it is contained in some line; in particular, it is homeomorphic to the standard triadic Cantor set. A theorem of Scěpin (see e.g. <cit.>) states that any homeomorphism of the standard triadic Cantor set C⊂ [0,1]⊂^2 extends to a homeomorphism  of ^2; moreover, this homeomorphism can be assumed to be the identity outside a large enough ball. In particular, we have: 


The homomorphism ϵ: (Σ_g) →(C) is onto. 



Moreover, (C) is a simple group <cit.>, and thus  the connected component of the identity in (C) is trivial. Therefore, ϵ descends to a continuous surjective homomorphism (using a slight abuse of notation)

    ϵ: (Σ_g) →(C),

where (Σ) has been endowed with the quotient topology coming from the compact-open topology on  (Σ_g). Observe that the kernel of this homeomorphism is exactly the pure  mapping class group (Σ_g), and thus we have a short exact sequence

    1 →(Σ_g) →(Σ_g) →(C) → 1


We are now going to give the version of the exact sequence (<ref>) when (Σ_g) is replaced by _g or _g. As we will see, the subgroup which appears on the right will be (isomorphic to) Thompson's group V. We start by giving a definition of this group. 



 §.§ Thompson's group V


Recall that Thompson's group V is the group of right-continuous bijections of the unit circle that map the set of dyadic rationals to itself, are differentiable except at finitely many points, and  on every interval of differentiability they are affine maps whose derivatives are powers of 2. The group V is well-known to be finitely presented, with respect to an explicit presentation. We refer the reader to the standard reference <cit.> for a thorough discussion on the different Thompson's groups. 
Extending results of <cit.>, we will prove: 


For every g≥ 0, there is a short exact sequence 
    1 →_c(Σ_g) →_g → V → 1.
  




As mentioned in the introduction, Proposition <ref> is shown in <cit.> in the case when g=0, where it is also shown that it splits over Thompson's group T; compare with Proposition <ref>.


In order to prove Proposition <ref>, it will be useful to work with a different incarnation of the group V (see <cit.> also). Namely, V is the group of self-transformations of the rooted 3-valent tree 𝒯 whose elements are encoded by equivalence classes of triples (T,T', σ), where T and T' are finite rooted subtrees of 𝒯 with the same number of leaves, and σ is a bijection between the set of leaves of T and T'. Such a triple extends to a transformation of 𝒯, and the equivalence relation responds to the fact that different triples may extend to the same transformation of 𝒯. 

Every element of V induces a homeomorphism of the space of ends of 𝒯, which is homeomorphic to the Cantor set C.  Thus V < (C). Armed with this alternate description, we adapt the arguments of <cit.> in order to prove Proposition <ref>


As mentioned above, the case g=0 is covered in <cit.>, so assume that g ≥ 1. Fix a rigid structure on Σ_g, and let 
c be the separating curve used to define it. The dual graph of the pants decomposition P underlying the rigid structure is naturally isomorphic to 𝒯, where the root corresponds to the unique pair of pants of Σ_g-P having c as boundary curve. 





We now define a homomorphism _g → V as follows. Let f∈_g and consider a defining subsurface Z for f.  We associate to f the triple (T,T',σ) ∈ V, where T (resp. T') is the  subtree of 𝒯 contained in Z (resp. f(Z)), and σ is the bijection between the sets of leaves of T and T', respectively, induced by the permutation between the sets of boundary components of Z and f(Z) given by f.
At this point, one easily checks that this correspondence gives rise to a well-defined surjective homomorphism _g → V.

We claim that the kernel of this homomorphism is exactly _c(Σ_g). Indeed, suppose f∈_g maps to the identity in V, and so in particular fixes every end of Σ_g. Consider a  defining subsurface Z for f.
 Since f does not permute ends we may assume, up to 
replacing Z by a suitably larger defining subsurface, that every component of Σ_g -Z is mapped to itself.

Therefore, after further enlarging Z if necessary, we deduce that f is the identity outside Z, and hence has compact support, as desired.






Next, observe that if we replace (Σ_g) by the asymptotic mapping class group _g in equation (<ref>), we obtain:

    1 →(Σ_g) ∩_g →_g →__g(C) → 1,

where __g(C) denotes the image of _g in (C) under the homomorphism ϵ of (<ref>). 

By the same argument that we used to determine the kernel of the homomorphism _g → V, we have: 


For every genus g, we have (Σ_g) ∩_g = _c(Σ_g).



At this point, the combination of equations (<ref>) and (<ref>) yields: 


 __g(C) is isomorphic to Thompson's group V. 


In particular, we have deduced that the restriction of the short exact sequence (<ref>) to _g is precisely the sequence (<ref>).




 §.§ A related Higman-Thompson group
 We now give a brief description of the subgroup V_2[ℤ_2] of (C) that appears when restricting the exact sequence (<ref>) to _g. The elements of V_2[ℤ_2] are transformations of 𝒯 encoded 
 by equivalence classes of tuples (T,T',σ,ε), where T, T' are subtrees, σ:∂ T→∂ T' is a bijection and ε∈ (/2)^∂ T'. 
 The group V_2[ℤ_2] is an example of the Higman-Thompson groups V_n[G], where n∈ℕ and G a subgroup of the symmetric group on n elements <cit.>. More concretely, it is associated to the subgroup G=_2 of permutations of n elements 
generated by the involution exchanging j and  n-j, for all j. (We remark that V = V_2[ Id] in this context.) 
Observe that we have an obvious inclusion V<V_2[ℤ_2]. However, by a surprising result of Bleak-Donoven-Jonušas <cit.> we have that, in fact, V ≅ V_2[ℤ_2], via an explicit element of (C) (a cellular automaton). 

Finally, we stress that V_2[ℤ_2] also appeared in <cit.> as the group of 
those homeomorphisms of a Cantor set embedded in 𝕊^2 which extend to smooth diffeomorphisms of the sphere.  
 
 
 Using the same reasoning as in the proof of Proposition <ref>, we have: 


For every g≥ 0, the restriction to the sequence (<ref>) to _g yields a short exact sequence 
    1 →_c(Σ_g) →_g → V_2[ℤ_2] → 1.



























§ FINITE PRESENTABILITY



In this section we prove Theorem <ref>. As mentioned in the introduction, the case g=0 was settled in <cit.>, and  will be a key ingredient in our proof:


The group _0 is finitely presented. 





It will be useful for us to give a brief description of the arguments used  in <cit.> for proving Theorem <ref>. The first ingredient, which will also play a central rôle here, is the following well-known result of Brown <cit.>: 



Let G be a group acting on a connected and simply-connected two-dimensional CW-complex X by permuting its cells. Suppose that:


  * The stabilizer of each vertex of X is finitely presented;

  * The stabilizer of every edge of X is finitely generated;

  * X/G is compact.

Then G is finitely presented. 



In <cit.>, Funar-Kapoudjian applied Theorem <ref> to the action of _0 on  a modification of the so-called pants complex <cit.>  of Σ_0. More concretely, they first consider a graph 𝒫_0 whose vertices are (isotopy classes of) pants decompositions of Σ_0 which differ from the pants decomposition underlying the rigid structure in a finite number of curves.
An edge of 𝒫_0 is given by two pants decompositions that are related by an elementary move, meaning that they differ in exactly two curves, which intersect exactly once (resp. twice) if their union fills a one-holed torus (resp. a four-holed sphere); see <cit.> for details. 

Using the same proof of the main result of Hatcher-Thurston <cit.>, one deduces that the graph 𝒫_0 is connected, and  that it becomes a  simply-connected 2-complex after gluing a 2-cell to every triangle, square, and pentagon of 𝒫_0. 
However, it turns out that the action of _0 on _0 is not cocompact, as there are infinitely many _0-orbits of squares <cit.>; in particular, Theorem <ref> cannot be applied to this situation. In order to overcome this, Funar-Kapoudjian construct a modification of 𝒫_0, called the reduced pants complex, by considering only two combinatorial types of squares, and show that 𝒫_0 is still simply connected. They then prove that _0 acts on 𝒫_0 satisfying all the hypotheses of Theorem <ref>, as desired. 

In fact, a minor variation of the arguments in <cit.> serves to prove the following strengthening of Theorem <ref>, which will be crucial for us: 



For every b∈∪{0}, the groups _0^b and _0^b are finitely presented.  



The definition of the reduced pants complex  from <cit.> makes sense also for Σ_0^b, with arbitrary b. By the same arguments as in <cit.>, this complex is connected and simply-connected and, using the same reasoning as in <cit.>, the groups _0^b and _0^b each act cocompactly on it. 
The stabilizer of every cell is  an extension of a finite permutation group by a finitely generated 
free-abelian group, and in particular finitely presented. At this point, the result follows from Theorem <ref>. 

 
We now turn to the proof of Theorem <ref>, whose statement we now recall: 

Theorem <ref>
The groups  _g and _g are finitely presented for every g≥ 0. 


In light of Theorem <ref>, it suffices to show the result for g positive. In order to do so, we will also use Brown's Theorem <ref>, this time using the action of _g (resp. _g) on the so-called cut-system complex 𝒦_g of Σ_g. We remark that cut-system complexes of finite-type surfaces were in fact used by Hatcher-Thurston <cit.> and Wajnryb <cit.> in order to compute finite presentations of mapping class groups. 

We now introduce the complex 𝒦_g. For concreteness, we choose to write the definitions for the surface Σ_g, although they make sense for an arbitrary connected orientable surface, of finite or infinite type. 

   A cut system of Σ_g is a set of g non-separating curves  whose union does not separate Σ_g. Let 𝒦_g be the simplicial graph whose vertices are cut systems on Σ_g, and where two cut systems are adjacent in 𝒦_g if they differ in exactly two curves, which intersect exactly once. 
   

Moreover, similar to the case of the pants complex, 𝒦_g will become a simply-connected 2-complex after gluing a 2-cell to certain circuits in 𝒦_g. Before explaining this, we need to borrow  some definitions from <cit.>.  A triangle in 𝒦_g consists of three pairwise-adjacent vertices of 𝒦_g; geometrically, a triangle corresponds to three  curves that pairwise intersect exactly once.  A square in 𝒦_g is a closed path with four vertices v_1, …,v_4 such that v_i and v_j are adjacent in 𝒦_g if and only if |i-j| =1  4; geometrically, a square corresponds to two elementary moves that occur in two disjoint one-holed tori. Finally, a pentagon in 𝒦_g consists of five vertices  v_1, …,v_5 such that v_i and v_j are adjacent in 𝒦_g if and only if |i-j| =1  5; geometrically, a pentagon is determined by five curves c_1, …, c_5 on Σ_g such that both c_i and c_i+1 belong to the cut system  v_i and i(c_i,c_i+2) =1, counting indices modulo 5. 

Armed with these definitions, we construct a 2-complex by gluing a 2-cell to every triangle, square, and pentagon of 𝒦_g. By a slight abuse of notation, we will denote the resulting complex by 𝒦_g also. The following result is essentially due to Hatcher-Thurston <cit.> and Wajnryb <cit.>:


For every g≥ 1, the complex 𝒦_g is connected and simply-connected. 




First, Hatcher-Thurston <cit.> and Wajnryb <cit.> proved that the cut-system complex of a finite-type surface is connected and simply-connected. To see that this is also the case for 𝒦_g observe that, for every finite set of vertices A ⊂𝒦_g, the union of the curves defining the elements of A together fill a finite-type subsurface of Σ_g. 


We are finally in a position to prove Theorem <ref>: 

 We prove the result for _g; the same argument, replacing every instance of  by , will give the result for _g. 
As mentioned earlier, we are going to apply Theorem <ref> to the action of _g on 𝒦_g. 

First of all, Theorem <ref> tells us that 𝒦_g is connected and simply-connected. Now, the classification theorem for infinite-type surfaces <cit.> and the fact that vertices are defined by a finite set of curves, together imply that _g acts transitively on the set of vertices of 𝒦_g. Since edges and 2-cells of 𝒦_g are defined in terms of intersection numbers, and by a finite set of curves, we deduce that _g acts cocompactly on 𝒦_g. 

Thus, it remains to justify why the stabiliser of a vertex (resp. edge) is finitely presented (resp. finitely generated). Consider first the case of the stabiliser of a vertex u  of 𝒦_g. Fix an orientation on each curve of u. Up to passing to a subgroup of finite index, and abusing notation, we may assume that every element of (u) fixes every curve of u together with its orientation. Now, cutting Σ_g open along the elements of u we obtain a surface homeomorphic to Σ_0^2g, and a short exact sequence 
    1→^g →(u) →_0^2g→ 1,
 where ^g is the group generated by the Dehn twists along the elements of u. Now, an extension of a finitely presented group by a finitely presented group is also 
finitely presented and thus Proposition  <ref> yields that (a finite-index subgroup of) (u) is finitely presented, as desired. 

Observe that the surface obtained from Σ_g by cutting the surface along the curves defining an edge of 𝒦_g is homeomorphic to Σ_0^2g-1. In light of this, as above we deduce that (a finite-index subgroup of) the stabiliser of an edge e∈𝒦_g fits in a short exact sequence 

    1→^g-1→(e) →^2g-1_0 → 1.
  
Therefore, the stabiliser of an edge is finitely presented, and in particular finitely generated. 
This finishes the proof of Theorem <ref>.
 


 


§ THE GROUP OF HALF-TWISTS IS DENSE


The goal of this final section is to prove Theorem <ref>, whose statement we now recall: 

Theorem <ref>
 _g is dense in (Σ_g). 



 In light of the result of Neretin and the second author <cit.> mentioned above, when g=0 this result may be interpreted as stating that homeomorphisms of the sphere minus a Cantor set may be approximated by diffeomorphisms. 


As it turns out, Theorem <ref> will be a consequence of a slightly stronger result, namely Theorem <ref> below. Before we state it, we need some preliminaries. Fix, once and for all, a rigid structure on Σ_g. Recall from section <ref> that the rigid structure is given by a separating curve on Σ_g, which will be denoted  c(∅)  for reasons that will become apparent below, plus a rigid structure on the planar component Σ_g^* of Σ_g - c(∅). Let P be the pants decomposition underlying the rigid structure on Σ_g^*. 

Similar to the situation  in subsection <ref>, there is an infinite rooted tree 𝒯' associated to P, whose vertices are the curves of P, with the root being c(∅). As such, every vertex of 𝒯'  is naturally labelled by a word w in the free semigroup F(L,R) generated by the two letters L (left) and R (right). We denote by c(w) the curve of P labelled by the word w∈ F(L,R), and write h(w) for the half-twist about c(w). 


In turn, this labelling induces a labelling of the set of pairs of pants of Σ_g^* by words in F(L,R). Indeed, we set P(w) to be the unique P-suited pair of pants of Σ_g^* that has c(w) as boundary component and is contained in the planar component of Σ_g  - c(w). 
 
 
Finally, set S(-∞) to be a fixed compact genus-g subsurface of Σ_g which contains c(∅). In addition, for each w ∈ F(L,R), let S(w) be a P-suited (therefore compact) subsurface whose interior is contained in Σ_g^* - c(w), and which has c(w) as a boundary component. Observe that the choice is far from unique; however, for any such choice, the set 
    {S(w) | w∈ F(L,R) ∪{-∞}}
 is proper, in the sense that every compact subsurface of Σ_g intersects only finitely  many elements of this set. Because of this, we will refer to any set {S(w) | w∈ F(L,R) ∪{-∞}} as above as a 
proper exhaustion by subsurfaces. 

After all this discussion, we have the following definition: 


A proper sequence of mapping classes  is a sequence {f_n}_n∈⊂(Σ_g), where 

    f_n = ∏_c w∈ F(L,R) ∪{-∞}
     |w| ≤ n  f(S(w)) h(w)^ϵ_w,
 where: 


  * f(S(w)) is an element of (S(w)),

  * |w| denotes the length of the word w,

  * ϵ_w ∈{-1,0,1} for all w,

  * The product is ordered lexicographically, and is defined from left to right.


Observe that if {f_n}_n∈⊂(Σ_g) is a proper sequence, then {f_n}_n∈⊂_g as well. In addition, since the defining subsurfaces of the f_n form a proper exhaustion of subsurfaces of Σ_g, we have:


If {f_n}_n∈⊂_g is a proper sequence, then it has a limit in (Σ_g). 



 Recall that (Σ_g) his equipped with the quotient topology coming from the compact-open topology on (Σ_g). We will prove: 


Every element of (Σ_g) is a limit of a proper sequence in _g. 



Observe that  Theorem <ref> obviously implies Theorem <ref>. 




Let f ∈(Σ_g) be an arbitrary mapping class. We will use an inductive argument, which we call a straightening of f, to produce a proper sequence of mapping classes whose limit is f. 

Consider c(∅). By the classification of infinite-type surfaces  <cit.>, there exist a P-suited genus-g subsurface S(-∞), and an element f_-∞ such that f_-∞  f sends c(∅) to itself. We say that f_-∞  f  straightens  c(∅).

Next, consider the curves c(L) and c(R). Again, there exists a P-suited planar subsurface S(∅) and an element f_∅∈(S(∅)) such that f_∅ f_-∞  f sends the set {c(L),c(R)} to itself. Therefore, up to precomposing with the half-twist h(∅) about c(∅) if necessary, we may in fact assume that c(L) (resp. c(R)) is sent to itself. We say that f_∅ f_-∞  f straightens the pair of pants P(∅). 


We continue this process inductively to find, for all n, a proper exhaustion by (compact) P-suited subsurfaces S(w) and elements f(S(w)) ∈(S(w)) such that, setting 
    f_n:=  ∏_c w∈ F(L,R) ∪{-∞}
     |w| ≤ n  f(S(w)) h(w)^ϵ_w,
 the mapping class 

    (f_n f)_n∈
  straightens every pair of pants P(w) for |w|≤ n. 

By Lemma <ref>, the sequence (f_n)_n∈ converges to an element f^*∈(Σ_g), in such way that f^* f fixes c(∅) and every curve of P. Moreover, by composing with an appropriate power of a Dehn twist at each step, we may assume that f^* f induces the trivial twist on each curve of P. Therefore, f^* f is an element of (S(-∞)). In other words, we have shown that, up to precomposing f_∅ with the inverse of this element, the sequence f_n converges to f in (Σ_g), as desired.

















§ STABLE HOMOLOGY
 

In this section, we adapt the methods of <cit.> to prove Theorem <ref>.
 First, observe that for n ≥ 1, gluing a pair of pants to a boundary component of S_g,n gives rise to an injective homomorphism 
    (S_g,n) →(S_g,n+1).
  In particular, one has a homomorphism 

    H_k((S_g,n),) → H_k((S_g,n+1),)

between the corresponding homology groups, which has a one-sided inverse coming from the capping homomorphism <cit.> and is therefore injective.   In <cit.>, Harer proved that the k-th homology group of (S_g,n) does not depend on n, provided k is sufficiently small with respect to g. The current best bound for what “sufficiently small" means is due to Boldsen <cit.> who, building up on unpublished work of Harer <cit.>, proved the following: : 
 
 Let g,n,k  ∈, with g≥ 0 and n, k≥ 1. Then, the homomorphism (<ref>) is an isomorphism for every k≤ 2g/3.


  
At this point, Lemma <ref> and Theorem <ref> together imply that 

    H_k(_c(Σ_g), ℤ) ≅ H_k((S_g,n),ℤ)
 
for all n≥ 1,  provided k ≤ 2g/3. For this reason, we will refer to the homology group H_k(_c(Σ_g), ℤ) as the stable  k-th homology group of the mapping class group of genus g. 

We are finally in a position to give a proof of Theorem <ref>, whose statement we now recall: 



Theorem <ref>
Let g≥ 1. For every k ≤ 2g/3, H_k(_g, ) (resp. H_k(_g, )) is isomorphic to the k-th stable homology group of the mapping class group of genus g. 

 
 
We adapt the proof of <cit.> to our setting. We treat the case of _g only, the other one being a direct translation. 

It is well-known that (S_g,n), where n≥ 1, is  torsion free and has a finite dimensional classifying space, which can be taken to be manifold (see e.g. <cit.>). 
Thus their integral homology is of finite type and by Harer's Stability Theorem 
 H_k(_c(Σ_g), ) is 
finitely generated for every k. 

Now, we apply the Lyndon-Hochschild-Serre spectral sequence method to
the short exact sequence (<ref>), which reads 

    1 →_c(Σ_g) →_g → V → 1.
 
In other words, there is a spectral sequence approximating the homology of _g whose second page is 

    E^2_pq= H_p(V,H_q(_c(Σ_g),)).
 
We now claim  that V acts trivially on H_q(_c(Σ_g), ) for all q.
Accepting this claim for the moment, it follows that the only non-zero 
terms of the spectral sequence above for the rational homology are  those with p=0, because V is acyclic (see <cit.>). 
In particular, the spectral sequence for the homology of _g collapses at the second page, which implies that 

    H_q(_g, )≅ H_q(_c(Σ_g),)
 for all q, as desired. Thus it remains to show:



Claim. V acts trivially on H_q(_c(Σ_g), ). 


Since H_k(_c(Σ_g), ) is a finitely generated abelian group, we get a finite-dimensional representation V →(N, ℚ). As V is simple, it follows that this representation is either trivial or injective. On the other hand, every finitely-generated subgroup of (N, ℚ) is residually finite, while V is not. In particular, the given representation is trivial.

This finishes the proof of Theorem <ref>.




By the sequence (<ref>), plus the fact that _c(Σ_g) (g≥ 3)
and V are both perfect, we obtain that _g and _g are perfect.  











§ RIGIDITY



The goal of this section is to prove Theorem <ref>. 
As mentioned above (G) will denote the automorphism group of the group G. 
 




 §.§ Automorphisms of _c(Σ_g)

The first ingredient in the proof of Theorem <ref> is the following: 


Suppose g≥ 0. Then (_c(Σ_g))= (Σ_g). 




We note that, prior to this work, this result had been obtained by Patel-Vlamis <cit.> for g≥ 4. 



Before explaining a proof of Proposition <ref>, we note the following easy observation:


_c(Σ_g) is a normal subgroup of (Σ_g) for every g≥ 0.




Let g∈_c(Σ_g)  and let Z be a  support for g. Consider an arbitrary h ∈(Σ_g), noting that any P-suited subsurface W containing h(Z) is a support for hgh^-1. Moreover, hgh^-1 induces the trivial permutation on the set of boundary components of W, and therefore hgh^-1∈_c(Σ).




In order to prove Proposition <ref>, we follow Ivanov's strategy for proving that automorphisms of (sufficiently complicated) finite-type mapping class groups are  conjugations. First, we observe:


Every automorphism of _c(Σ_g) sends  Dehn twists to Dehn twists. 

 

We stress that, if 4≤ g <∞, then Lemma <ref> quickly follows from <cit.> plus the fact that _c(Σ_g) is a direct limit of P-suited subsurfaces of Σ_g (Lemma <ref>). We now present a simpler argument, valid for arbitrary g, which was suggested to us by Juan Souto: 


Let ϕ∈(_c(Σ_g)). We take an arbitrary P-suited subsurface Z⊂Σ_g of genus g; in addition, we will assume that Z has at least seven boundary components (note that the latter assumption is only relevant if g<3). Since (Z) is finitely presented and _c(Σ_g) is a direct limit of pure mapping class groups of P-suited subsurfaces of genus g, by Lemma <ref>, we deduce that there exists a P-suited subsurface W⊂Σ_g such that ϕ((Z)) < (W). Let W^* be the (compact) subsurface of W supporting ϕ((Z)); in particular, this implies  that ϕ((Z)) < (W^*), and that (W^*) and ϕ((Z)) have the same centralizer in _c(Σ_g). 

Now, observe that, given a compact subsurface Y⊂Σ_g, the rank of the center of the centralizer of (Y) in _c(Σ_g) is equal to the number of boundary components of Y. In particular, since (Z) and ϕ((Z)) are isomorphic, the discussion of the previous paragraph implies that W^* and Z have the same number of boundary components. Again, <cit.> or <cit.> imply that they have equal genus, as Z has genus g and ϕ((Z)) < (W^*). In other words,  Z and W^* are homeomorphic. We claim:



Claim. The isomorphism ϕ: (Z) →(W^*) is induced by a homeomorphism Z→ W^*. 




If Z has genus ≥  2, then the claim is <cit.>. We now sketch an argument valid for arbitrary genus. 

The center C_Z of (Z) is the free abelian group generated by the Dehn twists along the boundary components of Z. Observe that ϕ induces an isomorphism 

    (Z) /C_Z →(W^*)/C_W^*,
 and that (Z) /C_Z is isomorphic to the pure mapping class group of the surface which results from gluing a punctured disk to every boundary component of Z. By the work of numerous authors (see, for instance, <cit.> for a comprehensive statement), the isomorphism  (<ref>) is induced by a homeomorphism f between the punctured surfaces. 
To see that f extends to a homeomorphism Z→ W^* inducing ϕ, we use the well-known lantern relation (see, for instance, <cit.>). Indeed, since we are assuming that Z has at least seven boundary components, given a boundary component a ⊂∂ Z we may find a collection b_1,…, b_6 of six curves, all of them essential in Z, such that the Dehn twist along a is expressed as a product of (suitable powers of) Dehn twists along the b_i's, via the lantern relation. 
Thus the claim follows. 




The claim above  implies that ϕ: (Z) →(W^*) is induced by a homeomorphism Z → W^*, and in particular sends Dehn twists to Dehn twists. 
The lemma now follows since Z was arbitrary and every curve on Σ_g is contained in some P-suited subsurface of genus g. 




Continuing with the arguments towards a proof of Proposition <ref>, we next claim that every automorphism of _c(Σ_g) induces an automorphism of the  
curve complex 𝒞(Σ_g); we refer the reader to the articles  <cit.> for various features of these complexes in the infinite-type setting.


Indeed, let
 ϕ : _c(Σ_g) →_c(Σ_g) be an automorphism. By Lemma <ref>, given a curve c ∈Σ_g there exists a curve d such that ϕ(T_c) = T_d, where T_a denotes the (left) Dehn twist about the curve a. Since T_a = T_b if and only if a = b, we get that the curve d above is in fact unique, and hence  ϕ induces a well-defined map 
    ϕ_* : (Σ_g) →(Σ_g)
 by the rule ϕ_*(c) = d.
 Observe that the map ϕ_* is simplicial, since two Dehn twists commute if and only if the defining curves are disjoint. Moreover, it is bijective, with inverse the simplicial self-map of (Σ_g) induced by ϕ^-1. 

Now we need the following analogue of Ivanov's celebrated result <cit.> on automorphisms of the curve complex: 



 The automorphism group of (Σ_g) is isomorphic to (Σ_g).



We are finally in a position to prove Proposition <ref>: 


We wish to show that the natural homomorphism 
    Λ: (Σ_g) →(_c(Σ_g))
 is an isomorphism. We first prove that Λ is surjective. To this end, let 
 ϕ∈(_c(Σ_g)), and consider the automorphism ϕ_* : (Σ_g) →(Σ_g) induced by ϕ. By Theorem <ref>, there exists a unique f ∈(Σ_g) with f(c) = ϕ_*(c) for all c ∈(Σ_g). 
 Note that ψ:=f^-1ϕ is an element of (_c(Σ_g)), and that  
 ψ_*(c) =c for all c ∈(Σ_g). 
 
Using again a direct translation of a classical argument in the finite-type setting, we will prove that the automorphism ψ is trivial, so that ϕ is in fact the conjugation by f. 
To this end, let h∈(Σ_g), and choose an arbitrary curve c on Σ_g. We have: 

    ψ(hT_c h^-1) = ψ(T_h(c)) = T_ψ_*(h(c))= T_h(c).

On the other hand: 

    ψ(hT_c h^-1) =ψ(h) ψ(T_c) ψ(h)^-1 =  ψ(h) T_ψ_*(c)ψ(h)^-1= T_ψ(h)(c)

Combining both equations, we obtain that h(c) = ψ(h)(c) for every curve c on Σ_g.   
By Theorem <ref> we deduce that h = ψ(h) for every h∈_c(Σ_g). In other words,  ψ is the identity on _c(Σ_g), as desired. 



 




 §.§ Proof of Theorem <ref>

We now embark in the proof of Theorem <ref>.  The key ingredient is the following unpublished result of Kapoudjian: 


Let  f ∈_g. Then f∈_c(Σ_g) if and only if for any finite set {f_i}⊂_g there exists h∈_g such that [h,f_iff_i^-1]=1 and
f belongs to  the normal closure of h in _g. 



Let f∈_g. Suppose first that f∈_c(Σ_g), and let {f_i}⊂_g be an arbitrary finite set. Choose a minimal  support Z for f, so that f(Z)=Z and f is identity outside Z.  
Observe that f_i(Z) is a  support for f_iff_i^-1. Then we may take h_0∈ such that h_0(Z)∩ f_i(Z)=∅ for all i, and set 
h=h_0fh_0^-1, which commutes with f_iff_i^-1. 

For the opposite direction let Z be a defining surface for f; recall this means that f sends Z to f(Z), and is rigid in the complement of Z ∪ f(Z). Up to enlarging Z if necessary, we may assume that Z has at
 least five boundary components. We now choose some finite set {f_i}⊂_g as in the statement, and which furthermore satisfies that the f_i all send Z to itself, and that the group generated by the f_i induces every possible permutation of the set of boundary components of Z. 


Seeking a contradiction, suppose that f∉_c(Σ_g). Then there exists a connected component of Σ_g-Z whose image by f is disjoint from itself. 
This implies that every defining surface of any h ∈_g as in the statement  must be contained in Z. Furthermore, the permutation of the set of boundary components of Z induced by h must commute with every element of the conjugacy class of the (nontrivial) permutation induced by f. 

Now, we claim that if a permutation σ in the symmetric group on k≥ 5 elements 
 commutes with every conjugate of a fixed nontrivial permutation σ_0, then σ is the trivial permutation. Indeed, σ commutes then with every element of the normal subgroup generated 
by σ_0; as k≥ 5 this normal subgroup contains the alternating group.  
Further, σ cannot be even as the alternating group is center-free. 
It follows that σ commutes with the even permutation τσ and hence σ commutes with τ, for any transposition τ. Then σ is in the center of the symmetric group which is trivial.



In particular,  the permutation induced by h is the trivial one, and h ∈_c(Σ_g). Hence f belongs to the normalizer of _c(Σ_g) in _g, contradicting our assumption. 


As an immediate consequence we get: 


_c(Σ_g) is a characteristic subgroup of _g.




The statement of Lemma <ref> and the proof are still valid 
when we replace _g by _g. 
This implies that _c(Σ_g) is  characteristic in _g also.
 

In  light of Corollary <ref>, every automorphism of _g (resp. _g) induces an automorphism of _c(Σ_g), which we have already determined in Proposition <ref>. Now, in order to calculate the automorphism group of _g and _g from this, we will make use of the following small technical result – this is surely well-known, but we include a proof for completeness. 


Let H be a normal subgroup of a group G, and suppose it has trivial centralizer in G. Suppose ψ: G→ G is an injective homomorphism of G such that ψ_| H=  id_H. Then ψ=   id_G. 




Since ψ is the identity on H and H is normal in G we have that, for every f∈ G and every h ∈ H, 
    fhf^-1= ψ(fhf^-1) = ψ(f) ψ(h) ψ(f^-1) =  ψ(f) h ψ(f^-1).
 Since h is arbitrary, it follows that ψ(f^-1)f belongs to the centralizer of H in G, and therefore ψ(f) = f, by hypothesis. Since f is also arbitrary, the result follows.



The motivation for Lemma <ref> is the following claim: 


The centralizer of _c(Σ_g) in (Σ_g) is trivial. 



 
Suppose f∈(Σ_g) commutes with every element of _c(Σ_g), and in particular with every Dehn twist. In particular, we have that 

    T_a= fT_a f^-1 = T_f(a),
 
for every curve a on Σ_g. But, as we mentioned in the proof of Proposition <ref>, the Alexander method of <cit.> implies that every element of (Σ_g) that fixes every curve on Σ_g is the identity.



We need the following definition before proving Theorem <ref>. Suppose Z is an orientable surface with non-empty boundary ∂ Z, where we assume that every connected component of ∂ Z has been parametrized by means of a map φ:∂ Z→ S^1, which is a homeomorphism on each component. 
A homeomorphism F:Z→ Z respects the boundary parametrization if  
φ∘ f|_∂ Z=φ. 
The parametrized mapping class group ^*(Z) is the 
group of isotopy classes of self-homeomorphisms of Z respecting the boundary parametrization, modulo 
isotopies which are the identity on the boundary. 
Observe that (Z) < ^*(Z) is a finite index normal subgroup and that, as a by-product of the definition, an element of ^*(Z) may induce a non-trivial permutation on the set of boundary components of Z. 


  We prove the result for _g, as the case of _g is dealt with along similar lines.
We first show that the natural homomorphism 
    N_(Σ_g)(_g) →(_g)
 is surjective. To this end, let
 ϕ: _g →_g be an automorphism. By Corollary <ref>, ϕ induces an automorphism  
    ϕ_c: _c(Σ_g) →_c(Σ_g),
 which  is the restriction of ϕ to _c(Σ_g).  Proposition <ref> implies that there exists Φ∈(Σ_g) such that 
 ϕ_c is conjugation by Φ, denoted    Ad_Φ. 
 
 From the proof of Lemma <ref> for any (compact) P-suited subsurface X⊂Σ_g of genus g, there  exists a compact subsurface Y=Φ(X)⊂Σ_g such that the restriction  ϕ_c|_(X) sends  (X) isomorphically onto (Y). 
 
Observe that for any P-suited subsurface X⊂Σ_g the group ^*(X) admits a canonical embedding into 
_g, by extending rigidly classes of homeomorphisms of X to classes of homeomorphisms of Σ_g. We claim:










 
 For any P-suited  subsurface X⊂Σ_g of genus g, the restriction 
 ϕ|_^*(X) sends  ^*(X) isomorphically onto  some copy of ^*(Φ(X)).
 Moreover,  ϕ|_^*(X) is conjugation by Φ. 
 
 
 
 As ϕ|_(X) is conjugation  by Φ, we have 
 ϕ(T_c)=T_Φ(c), when c∈∂ X is a boundary circle. 
 The stabilizer of the set {T_c | c∈∂ X} in _g is the stabilizer of the simplex of 𝒞(Σ_g) 
 determined by elements of ∂ X, with respect to the _g-action.   
 We call it the stabilizer of the multicurve. 
 
Let Y be a compact subsurface of Σ_g and denote by S_∂ Y the permutation group on the set of boundary components of Y. Looking at the induced permutation on the set of boundary components, we obtain canonical homomorphisms: 

    ^*(Y)→ S_∂ Y, ^*(Σ_g ∖(Y))→ S_∂ Y,

where (Y) denotes the interior of Y.  We denote by ^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)) the fibred product of the two 
 homomorphisms above, making the following diagram commutative: 
 
    [ ^*(Y)× _S_∂ Y^*(Σ_g ∖(Y))                         →              ^*(Σ_g ∖(Y));                         ↓                                                   ↓;                     ^*(Y)                         →                     S_∂ Y;                           ]

 Assume now that Y has genus g. 
 Then the stabilizer of the multicurve ∂ Y in (Σ_g) is 
 ^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)). 
 
 Further, the stabilizer in _g of the multicurve 
∂ X should be sent by ϕ into the stabilizer 
 of the multicurve ∂ Y, where Y=Φ(X). 
Since ^*(X) stabilizes ∂ X we derive that 
 ϕ(^*(X))⊂^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)).

 If π_1 and π_2 denote, respectively, the projections onto the first and second factors of the fibred product then 
 we derive a map π_1∘ϕ: ^*(X)→^*(Y), such that 
 π_1∘ϕ|_(X) is the conjugation Ad_Φ by Φ. 

We claim that  π_1∘ϕ=Ad_Φ. By composing with Ad_Φ^-1 
this reduces to show that if a homomorphism 
φ:^*(X)→^*(X)  restricts to identity on (X) then φ is the identity. If h∈^*(X) sends a boundary component a into b, then 

    T_b=φ(T_h(a))=φ(hT_ah^-1)=φ(h)T_aφ(h)^-1=T_φ(h)a

 so that φ(h)a=b. Thus the homomorphism S_∂ X→ S_∂ X 
induced by φ is identity. Then the Five Lemma  
implies that φ is injective and Lemma <ref> gives us the desired result, as the centralizer of (X) in ^*(X) is trivial, if the 
 complexity of the surface X is large enough. 


 
 Further, the image π_2∘ϕ((X)) is trivial, since ϕ((X))=(Y). 
 Therefore π_2∘ϕ factors through the quotient ^*(X)/(X), namely S_∂ X, and the map 
 S_∂ X→^*(Σ_g ∖(Y))→ S_∂ Y is the isomorphism induced by 
 the conjugation Ad_Φ. This implies that there exists a section 
 S_∂ Y→^*(Σ∖ Y) which determines an embedding of 
 ^*(Y) into (Σ_g) so that ϕ(^*(X)) is the image of ^*(Y). 
 
 


Continuing with the proof, observe that an immediate consequence of the proof of Theorem <ref> is that _g has a system of generators 𝒮 consisting  uniquely of elements belonging to ^*(X_i) for finitely many 
 P-suited subsurfaces X_i⊂Σ_g.  
 Lemma <ref> shows that the natural map ϕ_^*(X):^*(X)→(Σ_g) coincides with Ad_Φ|_^*(X). In particular, ϕ(s)=Ad_Φ(s), for 
 any s∈ S.  Since both ϕ and Ad_Φ are homomorphisms, we derive that ϕ(s)=Ad_Φ(s) 
for any  s∈_g. It follows that Ad_Φ(_g)⊆_g,  
 namely Φ∈ N_(Σ_g)(_g), as claimed.  



To see that the homomorphism N_(Σ_g)→(_g) is injective, suppose that Φ∈(Σ_g) induces the identity automorphism of _g. Then Φ also induces the identity automorphism of _c(Σ_g).
 At this point, Corollary <ref> tells us that  _c(Σ_g) is normal in _g, since it is characteristic. In addition, _c(Σ_g) has trivial centralizer in (Σ_g), and therefore in _g, by Lemma <ref>. Hence, Lemma <ref> implies that Φ is the identity. 




 §.§ A note on normalizers
  We now explore some  properties of the normalizer of _g (resp. _g) in (Σ_g); as we will see, these represent a certain departure from the case of finite-type mapping class groups. First, we have: 



 Let G be a subgroup of (Σ_g). Assume that   
 G∩(Σ_g) is a normal subgroup of (Σ_g). 
Then 
there is a natural surjective homomorphism
 
    N_(Σ_g)(G) → N_(C)(G/G∩(Σ_g)).

 
 

As G∩(Σ_g) is a normal subgroup of G we derive from the exact sequence (<ref>) that  G/G∩(Σ_g) is a subgroup of 
(C). Then there is a well-defined homomorphism  

    N_(Σ_g)(G) → N_(C)(G/G∩(Σ_g))

that sends every element H of the normalizer to its restriction to the space C  
of ends of Σ_g. 
Let now h∈ N_(C)(G/G∩(Σ_g)).  By the exact sequence 
(<ref>)  h admits a lift H∈(Σ_g). 
Conjugation by H sends the subgroup G∩(Σ_g) into itself, because it  
was assumed to be a normal subgroup of (Σ_g). 
It follows that H∈ N_(Σ_g)(G), so that the homomorphism in the statement is surjective. 


By a deep theorem of 
Rubin (see <cit.>), if G/G∩(Σ_g) is sufficiently large, in particular if 
G⊇_g, then there is an isomorphism 

    (G/G∩(Σ_g))≃ N_(C)(G/G∩(Σ_g)).


A recent result <cit.> shows that the outer automorphism group (V) of V is infinite. 
Combining this with the lemma above, we obtain: 

The homomorphisms (_g)→(V) and (_g)→(V) are surjective. In particular, 
(_g) and (_g) are infinite. 
 


From the exact sequences (<ref>) and (<ref>) the 
and Corollary <ref> _g∩(Σ_g)=_g∩(Σ_g)=_c(Σ_g). 
Then Lemma <ref>  shows that the assumptions of Lemma <ref> are satisfied. 
In particular we have surjective homomorphisms N_(Σ_g)(_g)→ N_(C)(V) 
and N_(Σ_g)(_g)→ N_(C)(V). By Rubin's Theorem  N_(C)(V)≃(V). 
On the other hand   N_(Σ_g)(_g)⊂(_g) and 
 N_(Σ_g)(_g)⊂(_g). 




We remark that a theorem of Ivanov <cit.> mentioned above asserts that ((S_g,n)) is always a finite group (in fact, trivial in all but finitely many cases). Thus Corollary <ref> represents a limitation in the dictionary between asymptotic and finite-type mapping class groups. 



Finally, we prove that the first homomorphism of Corollary <ref> has infinite kernel: 


For every g≥ 0, the homomorphism (_g)→(V) has infinite kernel. 

 As usual, we prove the result for _g only. 
Consider first the surface Σ_0^1, which recall that is homeomorphic to a sphere minus the union of a Cantor set and an isolated puncture. Using a totally analogous argument to that of Lemma <ref>, we deduce that there is a surjective homomorphism 

    (_0^1)→(V)

Recall the construction of the element t_∞ from Example <ref>. 
Let P be the pants decomposition underlying the rigid structure on Σ_0^1, and  t_∞∈(Σ_0^1) obtained as the product of all (say left) half-twists about the curves of P. 

We claim that t_∞∈ N_(Σ_0^1)(_0^1). To see this, observe that if Z is a defining surface  of an element φ∈_g, then each connected component of Σ_0^1-Z is preserved by t_∞φ t_∞^-1; moreover, this element acts as the identity on every such component. In particular, t_∞φ t_∞^-1∈_0^1 and thus t_∞∈ N_(Σ_0^1)(_0^1). Furthermore, the short exact sequence (<ref>) implies that t_∞^d∉_0^1 unless d=0, since  t_∞^d fixes every end of Σ_0^1 but  does not have compact support unless d=0. Hence it provides an example of an infinite order element in 
(_0^1). On the other hand, its image under the homomorphism (<ref>) is obviously trivial, and thus this homomorphism has infinite kernel.

After all this discussion, we treat the case of Σ_g. To do so, we simply embed Σ_0^1 in Σ_g in such way that the pants decomposition P underlying the rigid structure of Σ_0^1 is sent to the pants decomposition underlying the rigid structure of Σ_g. Using the same arguments as above, we deduce that the image of t_∞ under the homomorphism induced by the embedding Σ_0^1 →Σ_g produces an infinite-order element in the kernel of the homomorphism (_g)→(V).

























§ SURJECTIONS BETWEEN ASYMPTOTIC MAPPING CLASS GROUPS



In this section we will prove Theorem <ref>. 


Let 0≤ g<h and h≥ 2. Seeking a contradiction, suppose there were a surjective 
homomorphism  ϕ: _h →_g. 

For any f∈_c(Σ_h) there exists, by Lemma <ref>, 
some P-suited subsurface S_h,n⊂Σ_h such that f is in the image of 
(S_h,n) within _c(Σ_h). 
Let a∈ϕ(_c(Σ_h)), so that a=ϕ(f), for some f∈(S_h,n). 
For any finite set {a_i}⊂_g we can write a_i=ϕ(f_i), with 
f_i∈_h.  According to Lemma <ref>, 
there exists f'∈_h such that f belongs to 
the normal subgroup of _h generated by f' and such that [f',f_i ff_i^-1]=1.
This implies that a belongs to the normal subgroup of _g generated by 
b=ϕ(f') and [b,a_i a a_i^-1]=1. Lemma <ref> again implies that 
a∈_c(Σ_g). 

 

Restricting ϕ we obtain a homomorphism (S_h,n) →_c(Σ_g). Since (S_h,n) is finitely presented and _c(Σ_g) is a direct limit of finitely presented groups, by Lemma <ref>, we deduce that there is m∈ and a nontrivial map (S_h,n) →(S_g,m). But every such map has either 
a quotient of /10 as image, when h=2, or  trivial image, when h≥ 3, by <cit.> or <cit.>. 



Let now h≥ 3. By the above ϕ(f)=1, for any f∈(Σ_h), and therefore  
ϕ factors through V. Let ϕ':V→_g be the induced surjective homomorphism.  
If we compose the projection p:_g→ V with ϕ'  we obtain then 
a surjective homomorphism V→ V with nontrivial kernel. Since V is simple 
such a homomorphism should be trivial. This contradicts the surjectivity.  


Assume that h=2. The nested groups ϕ((S_2,n)) are quotients of /10 for 
every n, in particular the sequence is eventually constant and ϕ(_c(Σ_2)) 
is a quotient of /10. 
Therefore ϕ factors through the group Q=_2/[_c(Σ_2), _c(Σ_2)], yielding a surjective homomorphism ϕ':Q→_g.
Observe that the restriction of the sequence (<ref>) to Q reads 

    1→/10=H_1(_c(Σ_2))→ Q→ V→ 1.



 
Now, the normal subgroup K=ϕ'^-1(_c(Σ_g)) Q is infinite 
because it is the inverse image of an infinite subgroup
by a surjective homomorphism. Therefore K cannot be contained in the kernel of the projection homomorphism  p: Q→ V, which is finite. Since p(K) is a normal subgroup of V and V is simple,  
p(K)= V. As p is finite, we derive that Q/K must be finite.
On the other hand, K has infinite index in Q, because 
it is the preimage of the normal subgroup _c(Σ_g) of _g by the surjective 
homomorphism ϕ'. 
Hence we get  the desired contradiction. 




The same reasoning as above along with Remark <ref> 
implies that there are no surjective homomorphisms _h →_g (resp. _h →_g
and _h →_g) if g<h.
 







§ HOMOMORPHISMS FROM LATTICES AND PROPERTY (T)


We start this section by proving Theorem <ref>, which is a direct consequence of the existence of the short exact sequence (<ref>).

Before we prove the result, recall that a discrete group G is said to have Kazhdan's  Property (T) if every action of G by continuous affine isometries on a real Hilbert space has a fixed point. On the other hand,  G has the Haagerup property if it admits a proper action by continuous affine isometries on a Hilbert space. It follows that if G has both properties, then G is finite. We refer the reader to the book <cit.> for a thorough exposition of these properties. 
We may now prove Theorem <ref>, whose statement we now recall: 

Theorem <ref>
Let g≥ 0. Then _g and _g do not have Property (T). 




Recall from equation (<ref>) that there is a surjective homomorphism _g → V. It is known (see, for instance, <cit.>) that Property (T) is preserved under quotients. On the other hand, a result of Farley <cit.> asserts that V has the Haagerup property. Since V is infinite,  it follows that V does not have Property (T), and thus neither does _g.  The same argument yields the result for _g. 


We now proceed to give a proof of Theorem <ref>. The proof is again based on properties of the group V, plus 
a result of Farb-Masur <cit.> which asserts that every homomorphic image of a lattice in a finite-type mapping class group is finite. More concretely, they proved:



Let Γ be an irreducible lattice in a semisimple Lie group of real rank at least two. For every g, n≥ 0, every homomorphism Γ→(S_g,n) has finite image. 

 

Before giving a proof of Theorem <ref>, we remind the reader of its statement:


Theorem <ref>
Let Γ be a lattice in a  semisimple Lie group G of real rank at least two, where G has no compact factors isogenous to SU(1,n) or SO(1,n). 
Then, any homomorphism from Γ to _g (resp. _g) has finite image. 



Let G be a Lie group as in the statement. As such, G has Property (T) (see <cit.>, for instance), and therefore so does Γ (see <cit.>). For such a lattice Γ, let ϕ: Γ→_g be a homomorphism. Recall again the short exact sequence 
 (<ref>): 
    1→_c(Σ_g)→_gp→ V→ 1.
 
 As in the proof of Theorem <ref>, the fact that V has the Haagerup property implies that (p ∘ϕ)(Γ) is finite. But ϕ(Γ) ∩(p) is also finite. To see this, observe that since Γ is finitely presented (see <cit.>), Lemma <ref> implies that there exists a finite-type subsurface S of Σ_g such that ϕ(Γ) ∩(p) is contained in (S). At this point, Theorem <ref> tells us that ϕ(Γ) ∩(p) is also finite, as desired. 
 
 We obtain the result for _g in an analogous way. 
 






§ NON-LINEARITY


In this section we will prove Theorem <ref>. As mentioned in the introduction, we will do so by showing that _g contains a copy of Thompson's group F. Recall that F is the group of piecewise-linear self-homeomorphisms of [0,1] that preserve rational dyadic numbers, are differentiable except at finitely many dyadic rationals, and at each interval of differentiability the slopes are  powers of 2. 

A strongly related group which we will also need is Thompson's group T, whose definition is the same as that of F but with the unit circle 𝕊^1 instead of [0,1]. One has the well-known inclusions F⊂ T ⊂ V; for a proof, as well as a detailed discussion on Thompson's groups, see <cit.>. We now prove: 


Let g≥ 0. The short exact sequence (<ref>), that is 
    1 →_c(Σ_g) →_g → V → 1,
 splits over  Thompson's group F.
 
 
The result is known for g=0 (see <cit.>), where in addition it is shown that the sequence (<ref>) splits over  Thompson group's T, identified with the subgroup of _0 preserving the whole visible side of Σ_0.

Consider now a closed disk Σ_0,1 with a Cantor set removed from its interior. The surface Σ_0,1 comes equipped with an obvious rigid structure that comes from  that of Σ_0 under the natural subsurface embedding Σ_0,1↪Σ_0. 
The group of asymptotically rigid mapping classes of Σ_0,1 is the pure braided Thompson group 
BV, a subgroup of the braided Thompson group considered by Brin <cit.> and Dehornoy (<cit.>). 
When g>0 
the map (Σ_0,1)→(Σ_g) is injective. As a consequence BV injects into 
_g. On the other hand it is known that BV splits over F (see <cit.>). 


Finally, we prove Theorem <ref>:


 By Proposition <ref>, B_g contains an isomorphic copy of Thompson's group F. Now, F has simple commutator subgroup <cit.>, and in particular is not residually finite. Since finitely-generated subgroups of linear groups are residually finite and F is finitely generated, we deduce that _g (and thus _g) is not linear. 


































































































































































































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