We describe the automorphisms group of the complex of asymptotically trivial pants decompositions of a sphere punctured along the standard Cantor set and in particular find that it is not rigid.

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2000 MSC classsification: 57 N 05, 20 F 38, 57 M 07, 20 F 34

Keywords: universal mapping class group, pants complex, infinite type surfaces, group actions, braided Thompson group.

1 Definitions and statements

Inspired by Royden’s theorem on the holomorphic automorphisms of Teichmüller spaces Ivanov proved in [14] (subsequently completed by Korkmaz in [15]) that the automorphism group of the complex of curves of most compact surfaces coincides with the extended mapping class group. This was the start-point of many results of similar nature, coming under the page_seo_title of rigidity theorems. Margalit proved the rigidity (see [20]) of pants complexes and further work extended this to even stronger rigidity theorem (see e.g. [1, 21, 19, 11, 12, 13, 16] for a non-exhaustive list).

The study of such automorphisms groups in the pro-finite or pro-unipotent categories seems fundamental in Grothendieck’s program. For instance, although the pro-finite pants complexes are still rigid the automorphism group of the corresponding pro-finite curve complexes is a version of the Grothendieck-Teichmüller group ([18] and references there).

Simpler versions of this general question concern the solenoids, whose study was started in [2], and then infinite type surfaces corresponding to direct limits. The purpose of this article is to make progress in the second case using the formalism of asymptotically rigid homeomorphisms and braided Thompson groups developed in [9, 10]. A previous result in this direction is the rigidity theorem proved in [8] for an infinite type planar surface related to the Thompson group Tݑ‡Titalic_T (see [5]).

In this note we will consider an infinite surface obtained from the sphere by deleting the standard Cantor set from the equator, whose corresponding asymptotically rigid mapping class group is an extension ℬℬ\mathcalBcaligraphic_B (see [9]) of the Thompson group Vݑ‰Vitalic_V (see [5]) by an infinite pure mapping class group, which is closely related to the braided Thompson groups considered by Brin ([3]) and Dehornoy ([6, 7]). The novelty in the present setting is the appearance of some non-rigidity phenomenon of the corresponding pants complexes, and the page_content of the automorphisms group as a mapping class group.

1.2 The surface S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT

Let ݔ»2superscriptݔ»2\mathbbD^2blackboard_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT be the (hyperbolic) disc and suppose that its boundary ∂ݔ»2superscriptݔ»2\partial\mathbbD^2∂ blackboard_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is parametrized by the unit interval (with endpoints identified). Let τ*subscriptݜ\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT denote the (dyadic) Farey triangulation of ݔ»2superscriptݔ»2\mathbbD^2blackboard_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. This triangulation is given by the family of bi-infinite geodesics representing the standard dyadic intervals, i.e. the family of geodesics Iansuperscriptsubscriptݐ¼ݑŽݑ›I_a^nitalic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT joining the points p=a2nݑݑŽsuperscript2ݑ›p=\fraca2^nitalic_p = divide start_ARG italic_a end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG, q=a+12nݑžݑŽ1superscript2ݑ›q=\fraca+12^nitalic_q = divide start_ARG italic_a + 1 end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG on ∂ݔ»2superscriptݔ»2\partial\mathbbD^2∂ blackboard_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where a,nݑŽݑ›a,nitalic_a , italic_n are integers satisfying 0≤a≤2n-10ݑŽsuperscript2ݑ›10\leq a\leq 2^n-10 ≤ italic_a ≤ 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 1. Let T3subscriptݑ‡3T_3italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT be the dual graph of τ*subscriptݜ\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT, which is an infinite (unrooted) trivalent tree. Let ΣΣ\Sigmaroman_Σ be a closed δݛ¿\deltaitalic_δ-neighborhood of T3subscriptݑ‡3T_3italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.

Let S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT be the infinite surface obtained from gluing two copies of ΣΣ\Sigmaroman_Σ along its boundary. We assume in addition that the family of arcs coming from the two copies of τ*subscriptݜ\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT defines a collection of simple closed curves, denoted by Eݐ¸Eitalic_E.

A pants decomposition of the surface S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT is a maximal collection of distinct homotopically nontrivial simple closed curves on S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT which are pairwise disjoint and non-isotopic. The complementary regions (which are 3-holed spheres) are called pair of pants. The collection of simple closed curves Eݐ¸Eitalic_E is a pants decomposition of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. Subsequently, Eݐ¸Eitalic_E will be called the standard pants decomposition of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT.

A prerigid structure is a countable collection of disjoint line segments properly embedded into S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT, such that the complement of their union in S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT has two connected components.

A rigid structure is the data consisting of a pants decomposition and a prerigid structure such that :

1.
The traces of the prerigid structure on each pair of pants are made up of three connected components, called seams;

2.
Each pair of boundary circles of a given pair of pants, there is exactly one seam joining the two circles.

We arbitrarily fix a prerigid structure associated to the standard pants decomposition Eݐ¸Eitalic_E to obtain a rigid structure which is called the standard rigid structure of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. The complement in S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT of the union of lines of the canonical prerigid structure has two components: we distinguish one of them as the visible side of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. Remark that the visible side of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT is homeomorphic to the initial surface ΣΣ\Sigmaroman_Σ.

1.3 The mapping class group ℬℬ\mathcalBcaligraphic_B

A compact sub-surface Σ0,n⊂S0,∞subscriptnormal-Σ0ݑ›subscriptݑ†0\Sigma_0,n\subset S_0,\inftyroman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ⊂ italic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT (of genus zero with nݑ›nitalic_n boundary components) is admissible if its boundary is contained in the standard pants decomposition Eݐ¸Eitalic_E. The level of a compact sub-surface is the number nݑ›nitalic_n of its boundary components.

Let fݑ“fitalic_f be a homeomorphism of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. One says that fݑ“fitalic_f is rigid if it stabilizes the standard rigid structure, meaning that it maps the standard pants decomposition Eݐ¸Eitalic_E into itself, the seams into the seams, the visible side into the visible side.

Further, fݑ“fitalic_f is quasi-rigid if it stabilizes the standard pants decomposition Eݐ¸Eitalic_E.

Eventually, fݑ“fitalic_f is asymptotically rigid (resp. quasi-rigid) if there exists an admissible sub-surface Σ0,n⊂S0,∞subscriptnormal-Σ0ݑ›subscriptݑ†0\Sigma_0,n\subset S_0,\inftyroman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ⊂ italic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT such that f(Σ0,n)ݑ“subscriptnormal-Σ0ݑ›f(\Sigma_0,n)italic_f ( roman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ) is also admissible and the restriction of fݑ“fitalic_f on S0,∞∖Σ0,nsubscriptݑ†0subscriptnormal-Σ0ݑ›S_0,\infty\setminus\Sigma_0,nitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT ∖ roman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT is rigid (resp. quasi-rigid).

We denote by ℬℬ\mathcalBcaligraphic_B the group of orientation preserving isotopy classes of asymptotically rigid homeomorphisms of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT, which was called in [9] the universal mapping class group of genus zero.

Thompson groups and ℬℬ\mathcalBcaligraphic_B

There is a natural projection π:S0,∞→Σ:ݜ‹→subscriptݑ†0Σ\pi:S_0,\infty\rightarrow\Sigmaitalic_π : italic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT → roman_Σ such that the pullback of each arc of τ*subscriptݜ\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is the set of closed curves of Eݐ¸Eitalic_E. That lead us to define a bijection between the set of standard dyadic intervals and the set of closed curves of Eݐ¸Eitalic_E.

curves of E↔arcs of τ*↔standard dyadic intervals↔curves of ݐ¸arcs of subscriptݜ↔standard dyadic intervals\\textcurves of E\\leftrightarrow\\textarcs of \tau_{*}\% \leftrightarrow\\textstandard dyadic intervals\ curves of italic_E ↔ arcs of italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT ↔ standard dyadic intervals

Recall that the Thompson group Vݑ‰Vitalic_V is the group of right-continuous bijections of S1superscriptݑ†1S^1italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT that map images of dyadic rational numbers to images of dyadic rational numbers, that are differentiable except at finitely many images of dyadic rational numbers, and such that, on each maximal interval on which the function is differentiable, the function is linear with derivative a power of 2.

The Thompson group Tݑ‡Titalic_T is the group of piecewise linear homeomorphisms of S1superscriptݑ†1S^1italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT that map images of dyadic rational numbers to images of dyadic rational numbers, that are differentiable except at finitely many images of dyadic rational numbers and on intervals of differentiability on which the function is differentiable, the function is linear with derivative a power of 2.

For more details about Thompson groups, see [5]. Furthermore, we know that Tݑ‡Titalic_T can be viewed as an asymptotic mapping class group of the planar surface ΣΣ\Sigmaroman_Σ.

Let K∞*superscriptsubscriptݐ¾K_\infty^{*}italic_K start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT be the an inductive limit ⋃nK*(3×2n)subscriptݑ›superscriptݐ¾3superscript2ݑ›\bigcup_nK^{*}(3\times 2^n)⋃ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( 3 × 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) where K*(n)superscriptݐ¾ݑ›K^{*}(n)italic_K start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_n ) is the pure mapping class group of the nݑ›nitalic_n-holed sphere. We have the exact sequence ([9]) :

1⟶K∞*⟶ℬ⟶V⟶1⟶1superscriptsubscriptݐ¾⟶ℬ⟶ݑ‰⟶11\longrightarrow K_\infty^{*}\longrightarrow\mathcalB\longrightarrow V\longrightarrow 11 ⟶ italic_K start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ⟶ caligraphic_B ⟶ italic_V ⟶ 1

Moreover, we have the following result:

Proposition 1 ([9]).

The Thompson group Tݑ‡Titalic_T is the subgroup of elements of ℬℬ\mathcalBcaligraphic_B which preserve the visible side of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT.

Generators of ℬℬ\mathcalBcaligraphic_B

The group ℬℬ\mathcalBcaligraphic_B is generated by the twist tݑ¡titalic_t (see Fig 2), the braid πݜ‹\piitalic_π (Fig 3 ) and the two generators αݛ¼\alphaitalic_α and βݛ½\betaitalic_β of Tݑ‡Titalic_T. For more details about the definition of these generators, see section 3 of [9].

Some subgroups of ℬℬ\mathcalBcaligraphic_B

Proposition 2.

The subgroup of ℬℬ\mathcalBcaligraphic_B consisting of those mapping classes represented by rigid homeomorphisms is isomorphic to PSL2(ℤ)ݑƒݑ†subscriptݐ¿2ℤPSL_2(\mathbbZ)italic_P italic_S italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_Z ).

Let Σ0,nsubscriptΣ0ݑ›\Sigma_0,nroman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT be a nݑ›nitalic_n-holed sphere embedded in S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT such that its boundary components lie in the standard decomposition Eݐ¸Eitalic_E of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. Recall that the mapping class group ℳ(Σ0,n)ℳsubscriptΣ0ݑ›\mathcalM(\Sigma_0,n)caligraphic_M ( roman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ) is the group of isotopy classes of homeomorphisms of Σ0,nsubscriptΣ0ݑ›\Sigma_0,nroman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT preserving the orientation. The elements of ℳ(Σ0,n)ℳsubscriptΣ0ݑ›\mathcalM(\Sigma_0,n)caligraphic_M ( roman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ) are represented by homeomorphisms which can permute the boundary components of Σ0,nsubscriptΣ0ݑ›\Sigma_0,nroman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT. We can assume that they also preserve the trace of these boundary components on the visible side. Then there exists an embedding ℳ(Σ0,n)→ℬ→ℳsubscriptΣ0ݑ›ℬ\mathcalM(\Sigma_0,n)\rightarrow\mathcalBcaligraphic_M ( roman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ) → caligraphic_B obtained by extending rigidly a homeomorphism representing a mapping class of ℳ(Σ0,n)ℳsubscriptΣ0ݑ›\mathcalM(\Sigma_0,n)caligraphic_M ( roman_Σ start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ).

1.4 The extended mapping class group ℬ12^^superscriptℬ12\widehat\mathcalB^\frac12over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG

We now define some mapping classes which are not represented by asymptotically rigid homeomorphisms. These elements will generate the extended mapping class group ℬ12^^superscriptℬ12\widehat\mathcalB^\frac12over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG.

1.4.1 The symmetry

Let iRsubscriptݑ–ݑ…i_Ritalic_i start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT be the isotopy class of the quasi-rigid homeomorphism acting as the symmetry on S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT which permute the visible side and the invisible side on each pair of pants of the standard decomposition Eݐ¸Eitalic_E.

1.4.2 The half-twists

Let a,b,cݑŽݑݑa,b,citalic_a , italic_b , italic_c three curves of the standard decomposition Eݐ¸Eitalic_E bounding a pair of pants denoted by PݑƒPitalic_P. By cutting along each of these three curves, we define three infinite connected components of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT respectively denoted by Sa,Sb,Scsubscriptݑ†ݑŽsubscriptݑ†ݑsubscriptݑ†ݑS_a,S_b,S_citalic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and one pair of pants Sa,b,csubscriptݑ†ݑŽݑݑS_a,b,citalic_S start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT. We ask that the trace of bݑbitalic_b on the visible side is sent by Da,b,csubscriptݐ·ݑŽݑݑD_a,b,citalic_D start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT to the trace of cݑcitalic_c on the invisible side, and the trace of cݑcitalic_c on the visible side is sent onto the trace of bݑbitalic_b on the invisible side. Then we quasi-rigidly extend this homeomorphism on S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. This means that Da,b,csubscriptݐ·ݑŽݑݑD_a,b,citalic_D start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT acts on Sasubscriptݑ†ݑŽS_aitalic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT as the identity, sending the visible side of Sbsubscriptݑ†ݑS_bitalic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT on the invisible side of Scsubscriptݑ†ݑS_citalic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, the invisible side of Sbsubscriptݑ†ݑS_bitalic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT on the visible side of Scsubscriptݑ†ݑS_citalic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, the visible side of Scsubscriptݑ†ݑS_citalic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT on the invisible side of Sbsubscriptݑ†ݑS_bitalic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, the invisible side of Scsubscriptݑ†ݑS_citalic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT on the visible side of Sbsubscriptݑ†ݑS_bitalic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT. We denote by da,b,csubscriptݑ‘ݑŽݑݑd_a,b,citalic_d start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT the isotopy class of the homeomorphism Da,b,csubscriptݐ·ݑŽݑݑD_a,b,citalic_D start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT and call it the half-twist along aݑŽaitalic_a by respect to bݑbitalic_b and cݑcitalic_c. This choice does not depend on the choice of the curves a,b,cݑŽݑݑa,b,citalic_a , italic_b , italic_c in their homotopy classes. We denote by supp(da,b,c)ݑ ݑ¢ݑݑsubscriptݑ‘ݑŽݑݑsupp(d_a,b,c)italic_s italic_u italic_p italic_p ( italic_d start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT ) the sub-surface of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT on which Da,b,csubscriptݐ·ݑŽݑݑD_a,b,citalic_D start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT acts non trivially up to homotopy. With the previous notations, we have supp(da,b,c)=Sa,b,c∪Sb∪Scݑ ݑ¢ݑݑsubscriptݑ‘ݑŽݑݑsubscriptݑ†ݑŽݑݑsubscriptݑ†ݑsubscriptݑ†ݑsupp(d_a,b,c)=S_a,b,c\cup S_b\cup S_citalic_s italic_u italic_p italic_p ( italic_d start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT ∪ italic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∪ italic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT.

Note that each curve of the standard decomposition Eݐ¸Eitalic_E is associated to a standard dyadic interval of the form Ikn:=[k2n,k+12n]assignsuperscriptsubscriptݐ¼ݑ˜ݑ›ݑ˜superscript2ݑ›ݑ˜1superscript2ݑ›I_k^n:=[\frack2^n,\frack+12^n]italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT := [ divide start_ARG italic_k end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG , divide start_ARG italic_k + 1 end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG ] where kݑ˜kitalic_k and nݑ›nitalic_n are integers such that 0≤k≤2n-10ݑ˜superscript2ݑ›10\leq k\leq 2^n-10 ≤ italic_k ≤ 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 1. Moreover, the standard dyadic intervals I2kn+1superscriptsubscriptݐ¼2ݑ˜ݑ›1I_2k^n+1italic_I start_POSTSUBSCRIPT 2 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT and I2k+1n+1superscriptsubscriptݐ¼2ݑ˜1ݑ›1I_2k+1^n+1italic_I start_POSTSUBSCRIPT 2 italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT both define with Iknsuperscriptsubscriptݐ¼ݑ˜ݑ›I_k^nitalic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT a pair of pants in Eݐ¸Eitalic_E. Hence, each half-twist along a curve belonging in Eݐ¸Eitalic_E is defined by a couple (k,n)∈ℕ2ݑ˜ݑ›superscriptℕ2(k,n)\in\mathbbN^2( italic_k , italic_n ) ∈ blackboard_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT such that 0≤k≤2n-10ݑ˜superscript2ݑ›10\leq k\leq 2^n-10 ≤ italic_k ≤ 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 1 and we will use the notation

da,b,c=da=dIknsubscriptݑ‘ݑŽݑݑsubscriptݑ‘ݑŽsubscriptݑ‘superscriptsubscriptݐ¼ݑ˜ݑ›d_a,b,c=d_a=d_I_k^nitalic_d start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT

For all n≤mݑ›ݑšn\leq mitalic_n ≤ italic_m and j,kݑ—ݑ˜j,kitalic_j , italic_k satisfying 0≤j≤2n-10ݑ—superscript2ݑ›10\leq j\leq 2^n-10 ≤ italic_j ≤ 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 1 and 0≤k≤2n-10ݑ˜superscript2ݑ›10\leq k\leq 2^n-10 ≤ italic_k ≤ 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 1, we have supp(dIjn)∩supp(dIkm)=∅ݑ ݑ¢ݑݑsubscriptݑ‘superscriptsubscriptݐ¼ݑ—ݑ›ݑ ݑ¢ݑݑsubscriptݑ‘superscriptsubscriptݐ¼ݑ˜ݑšsupp(d_I_j^n)\cap supp(d_I_k^m)=\emptysetitalic_s italic_u italic_p italic_p ( italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ∩ italic_s italic_u italic_p italic_p ( italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = ∅ or supp(dIjn)⊂supp(dIkm)ݑ ݑ¢ݑݑsubscriptݑ‘superscriptsubscriptݐ¼ݑ—ݑ›ݑ ݑ¢ݑݑsubscriptݑ‘superscriptsubscriptݐ¼ݑ˜ݑšsupp(d_I_j^n)\subset supp(d_I_k^m)italic_s italic_u italic_p italic_p ( italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ⊂ italic_s italic_u italic_p italic_p ( italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ). Therefore, for all sequence (ji,ni,pi)i∈ℕsubscriptsubscriptݑ—ݑ–subscriptݑ›ݑ–subscriptݑݑ–ݑ–ℕ(j_i,n_i,p_i)_i\in\mathbbN( italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ∈ blackboard_N end_POSTSUBSCRIPT of elements of ℕ×ℕ×ℤℕℕℤ\mathbbN\times\mathbbN\times\mathbbZblackboard_N × blackboard_N × blackboard_Z such that for all i∈ℕݑ–ℕi\in\mathbbNitalic_i ∈ blackboard_N, 0≤ni≤2ni-10subscriptݑ›ݑ–superscript2subscriptݑ›ݑ–10\leq n_i\leq 2^n_i-10 ≤ italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - 1, the sequence (ni)i∈ℕsubscriptsubscriptݑ›ݑ–ݑ–ℕ(n_i)_i\in\mathbbN( italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ∈ blackboard_N end_POSTSUBSCRIPT is non decreasing, so we can define the infinite product ∏i=0∞(dIjini)pisuperscriptsubscriptproductݑ–0superscriptsubscriptݑ‘superscriptsubscriptݐ¼subscriptݑ—ݑ–subscriptݑ›ݑ–subscriptݑݑ–\prod_i=0^\infty(d_I_j_i^n_i)^p_i∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT as an element of the mapping class group of the surface S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT.

1.4.3 The group ℬ12superscriptℬ12\mathcalB^\frac12caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT

The group ℬ12superscriptℬ12\mathcalB^\frac12caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT is the group generated by the half-twists around curves of the standard decomposition Eݐ¸Eitalic_E and the generators αݛ¼\alphaitalic_α and βݛ½\betaitalic_β of the Thompson group Tݑ‡Titalic_T.

Remark that the braid and the twist are both generated by some half-twists. Indeed, a twist along a curve cݑcitalic_c is obtained by iterating twice a half-twist along the curve cݑcitalic_c. Moreover, a braid on a pair of pants bounded by three curves a,b,cݑŽݑݑa,b,citalic_a , italic_b , italic_c comes from the composition of the half-twists along each of these three curves. Finally, note that a half-twist along any curve of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT is conjugated to a half-twist along a curve of Eݐ¸Eitalic_E by an element of ℬℬ\mathcalBcaligraphic_B, since ℬℬ\mathcalBcaligraphic_B acts transitively on simple closed curves of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT. In short, we have the following relations, where f∈ℬݑ“ℬf\in\mathcalBitalic_f ∈ caligraphic_B :

tc=dc2subscriptݑ¡ݑsuperscriptsubscriptݑ‘ݑ2t_c=d_c^2italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

π=dadb-1dc-1ݜ‹subscriptݑ‘ݑŽsuperscriptsubscriptݑ‘ݑ1superscriptsubscriptݑ‘ݑ1\pi=d_ad_b^-1d_c^-1italic_π = italic_d start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT

1.4.4 Definition of ℬ12^^superscriptℬ12\widehat\mathcalB^\frac12over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG

The group ℬ12^normal-^superscriptℬ12\widehat\mathcalB^\frac12over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG is the group of isotopy classes of asymptotically quasi-rigid homeomorphisms of the surface S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT.

Since half-twists are quasi-rigid homeomorphisms of S0,∞subscriptݑ†0S_0,\inftyitalic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT, we see that ℬ12^^superscriptℬ12\widehat\mathcalB^\frac12over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG contains ℬ12superscriptℬ12\mathcalB^\frac12caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT as a subgroup. Also, we note that the symmetry iRsubscriptݑ–ݑ…i_Ritalic_i start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is an element of ℬ12^^superscriptℬ12\widehat\mathcalB^\frac12over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG and that iRsubscriptݑ–ݑ…i_Ritalic_i start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is an infinite product of all half-twists around curves of Eݐ¸Eitalic_E :

iR=∏ci∈Edcisubscriptݑ–ݑ…subscriptproductsubscriptݑݑ–ݐ¸subscriptݑ‘subscriptݑݑ–i_R=\prod_c_i\in Ed_c_iitalic_i start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT = ∏ start_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_E end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT

Let x∈ℬ12^ݑ¥^superscriptℬ12x\in\widehat\mathcalB^\frac12italic_x ∈ over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG. Then there exists an admissible sub-surface S⊂S0,∞ݑ†subscriptݑ†0S\subset S_0,\inftyitalic_S ⊂ italic_S start_POSTSUBSCRIPT 0 , ∞ end_POSTSUBSCRIPT outside of which xݑ¥xitalic_x stabilizes the decomposition Eݐ¸Eitalic_E. We can assume Sݑ†Sitalic_S is bounded by a set of curves of type Ijn0,0≤j≤2n0-1superscriptsubscriptݐ¼ݑ—subscriptݑ›00ݑ—superscript2subscriptݑ›01\I_j^n_0,0\leq j\leq 2^n_0-1\ italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , 0 ≤ italic_j ≤ 2 start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - 1 , where n0∈ℕsubscriptݑ›0ℕn_0\in\mathbbNitalic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N. Therefore, any element x∈ℬ12^ݑ¥^superscriptℬ12x\in\widehat\mathcalB^\frac12italic_x ∈ over^ start_ARG caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_ARG can be written as an infinite product

x=f⋅∏i=0∞(dIaini)piݑ¥⋅ݑ“superscriptsubscriptproductݑ–0superscriptsubscriptݑ‘superscriptsubscriptݐ¼subscriptݑŽݑ–subscriptݑ›ݑ–subscriptݑݑ–x=f\cdot\prod_i=0^\infty(d_I_a_i^n_i)^p_iitalic_x = italic_f ⋅ ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT
where f∈ℬ12ݑ“superscriptℬ12f\in\mathcalB^\frac12italic_f ∈ caligraphic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT quasi-rigidly acts outside Sݑ†Sitalic_S, (ni)i≥1subscriptsubscriptݑ›ݑ–ݑ–1(n_i)_i\geq 1( italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ≥ 1 end_POSTSUBSCRIPT is an eventually increasing (actually there are no more than 2nsuperscript2ݑ›2^n2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT occurences of nݑ›nitalic_n) sequence of natural numbers, 0≤ai≤2ni-10subscriptݑŽݑ–superscript2subscriptݑ›ݑ–10\leq a_i\leq 2^n_i-10 ≤ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - 1 and pi∈ℤsubscriptݑݑ–ℤp_i\in\mathbbZitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_Z for all i∈ℕݑ–ℕi\in\mathbbNitalic_i ∈ blackboard_N. Moreover, half-twists dIainisubscriptݑ‘superscriptsubscriptݐ¼subscriptݑŽݑ–subscriptݑ›ݑ–d_I_a_i^n_iitalic_d start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT act as the identity in Sݑ†Sitalic_S. Hence, fݑ“fitalic_f commutes with ∏i=0∞(dIaini)pisuperscriptsubscriptproductݑ–0superscriptsubscriptݑ‘superscriptsubscriptݐ¼subscriptݑŽݑ–subscriptݑ›ݑ–s

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