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# Surface S0,∞ with its canonical rigid structure

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We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichm\"ul...

## Context in source publication

Context 1
... choice of a reference rigid structure defines the canonical rigid structure (cf. Figure 6). The dyadic regular (unrooted) tree T * is embedded onto the visible side of S 0,∞ , as the dual tree to the canonical decomposition (into hexagons). ...

## Citations

... By the above identification, the group V can be regarded as a group acting on the Cantor set C which is an alternative interpretation of the boundary of the rooted infinite binary tree denoted by ∂(T ∞ ). The group V can then be interpreted as a group acting on the infinite binary tree by partial automorphism of the rooted infinite binary tree T ∞ which resembles the definition give by Funar and Sergiescu's description in [14]. ...
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Motivated by Burillo, Cleary and Roever's summary on obstructions of subgroups of Thompson's group $V,$ we explored the higher dimensional version of the groups, Brin-Thompson groups $nV$ and $SV,$ a class of infinite dimensional Brin-Thompson groups and an easy class of the twisted version of the Brin-Thompson groups $SV_G$ with some certain condition. We found that they have similar obstructions as Thompson's group $V$ on the torsion subgroups and a selection of the interesting Baumslag-Solitor groups are excluded as the subgroups of $SV$ and $SV_G.$ We also discuss a little on the group $\mathscr{S}V,$ an even larger class relaxing some of the "finiteness condition" and observe that some of the restrictions on subgroups disappear.
... On the other hand, the analogy between PPSL 2 (Z) and the mapping class group suggests an extension of the former by an infinite symmetric group, or more generally by an infinite braid group, which is still a discrete subgroup of Homeo + (S 1 ), cf. [14] and the references therein. The reduction of H by the additional symmetry yields the space , or heuristically S ∞ V , which is naturally identified with the free bsosonic field, one of the simplest CFT 2 . ...
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We review and reformulate old and prove new results about the triad , which provides a universal generalization of the classical automorphic triad . The leading P or p in the universal setting stands for piecewise, and the group $$\mathrm{PPSL}_2({\mathbb Z})$$ plays at once the role of universal modular group, universal mapping class group, Thompson group T and Ptolemy group. In particular, we construct and study new framed holographic coordinates on the universal Teichmüller space and its symmetry group $$\mathrm{PPSL}_2({{\mathbb {R}}})$$, the group of piecewise $$\mathrm{PSL}_2({{\mathbb {R}}})$$ homeomorphisms of the circle with finitely many pieces, which is dense in the group of orientation-preserving homeomorphisms of the circle. We produce a new basis of its Lie algebra $$ppsl_2({{\mathbb {R}}})$$ and compute the structure constants of the Lie bracket in this basis. We define a central extension of $$ppsl_2({{\mathbb {R}}})$$ and compare it with the Weil-Petersson form. Finally, we construct a $$\mathrm{PPSL}_2({{\mathbb {Z}}})$$-invariant 1-form on the universal Teichmüller space formally as the Maurer-Cartan form of $$ppsl_2({{\mathbb {R}}})$$, which suggests the full program for developing the theory of automorphic functions for the universal triad which is analogous, as much as possible, to the classical triad. In the last section we discuss the representation theory of the Lie algebra $$ppsl_2({{\mathbb {R}}})$$ and then pursue the universal analogy for the invariant 1-form $$E_2(z)dz$$, which gives rise to the spin 1 representation of $$psl_2({{\mathbb {R}}})$$ extended by the trivial representation. We conjecture that the corresponding automorphic representation of $$ppsl_2({{\mathbb {R}}})$$ yields the bosonic CFT$$_2$$. Relaxing the automorphic condition from $$\mathrm{PSL}_2({{\mathbb {Z}}})$$ to its commutant allows the increase of the space of 1-forms six-fold additively in the classical case and twelve-fold multiplicatively in our universal case. This leads to our ultimate conjecture that we can realize the Monster CFT$$_2$$ via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichmüller and the Monster CFT$$_2$$ theories to the three-dimensional quantum gravity.
... In this article, we pursue the study initiated in [GLU20] of a particular family of braided Thompson-like groups. Our framework, largely inspired by [FK08] (see the survey [FKS12] and the references therein for more background), is the following. Fix a locally finite tree A embedded into the plane in such a way that its vertex-set is closed and discrete. ...
Preprint
In this article, we introduce a new family of groups, called Chambord groups and constructed from braided strand diagrams associated to specific semigroup presentations. It includes the asymptotically rigid mapping class groups previously studied by the authors such as the braided Higman-Thompson groups and the braided Houghton groups. Our main result shows that polycyclic subgroups in Chambord groups are virtually abelian and undistorted.
... On the other hand, the map from a finite binary tree to a different finite binary tree with the same number of leaves can be regarded as partial automorphism of the infinite binary tree [18]. Since the Cantor set can be identified with the boundary of the infinite binary tree, an element of the Thompson's group V can be regarded as the homeomorphism of the Cantor set that preserves finite number of the partitions of the Cantor set. ...
... This group can also be regarded as the central extension of the original Thompson's group V by the "stable braid groups " which includes all braid groups [18] by V . Similar to the case in F , T , V and the Brown-Thompson's groups. ...
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Golan and Sapir \cite{MR3978542} proved that the Thompson's groups $F$, $T$ and $V$ have linear divergence. In the current paper, we focus on the divergence properties of several generalisation of the Thompson's groups, we first consider the Brown-Thompson's groups $F_n$, $T_n$ and $V_n$ and found out that these groups also have linear divergence function. We then consider the braided Thompson's groups $BF$ and $\widehat{BF}$ and $\widehat{BV}$ together with the result in \cite{Kodama:2020to} we conclude that theses groups have linear divergence.
... In fact, some of the tools in Grothendick's theory are profinite versions of notions discovered by Thurston. The interested reader may refer to the surveys [84,235]. Conversely, Grothendieck's dessins d'enfants were studied by several authors in the setting of Thurston's theory; we refer to the surveys [101,102]. ...
Preprint
We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book In the tradition of Thurston: Geometry and topology'', edited by K. Ohshika and A. Papadopoulos (Springer, 2020).
... Here BV is the braided version of V introduced independently by Brin [7] and Dehornoy [17] in 2006. This group has intriguing connections to Teichmüller theory and the study of big mapping class groups [19]. ...
Preprint
We prove that the lift of Thompson's group $T$ to the real line contains the additive group $\mathbb{Q}$ of the rational numbers. This gives an explicit, natural example of a finitely presented group that contains $\mathbb{Q}$, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe. We also prove that $\mathbb{Q}$ can be embedded into a finitely presented simple group. Specifically, we describe a simple group $T\!\hspace{0.1em}\mathcal{A}$ of homeomorphisms of the circle that contains $\mathbb{Q}$, and we prove that $T\!\hspace{0.1em}\mathcal{A}$ is two-generated and has type $\mathrm{F}_\infty$. Our method for proving finiteness properties extends existing techniques to allow for groups of homeomorphisms that are locally more complicated than the standard Thompson groups.
... A positive answer to the above question, in the case of n = 0, is conjectured in [48, p. 967]. The question of whether B ∞ is finitely presented appears in [49]. ...
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We survey recent developments on mapping class groups of surfaces of infinite topological type.
... The corresponding cohomology classes e(T * ab ) and e(T ♯ ab ) in H 2 (T ; Z) are equal to 12e and 21e respectively, where e ∈ H 2 (T ; Z) is the Euler class (see Funar-Sergiescu [16] and Funar-Kapoudjian-Sergiescu [15]). Thus T * ab is isomorphic to T 12 and T ♯ ab is isomorphic to T 21 . ...
Preprint
In this note, we show that the sets of all stable commutator lengths in the braided Ptolemy-Thompson groups are equal to non-negative rational numbers.
... One of the reasons why Thompson's groups have been studied so much is that they appear in various guises throughout mathematics. Invented around 1965 in the context of mathematical logic and (un)solvable word problems (see [MT73] and [Tho80]), they have found interpretations that range from universal algebra, category theory, homotopy and shape theory (see, for example, [HH82], [BG84], [FH93], and [FL10]), to the geometry of piecewise-linear maps, Teichmüller theory, mapping class groups, and braids (see the expositions in [FKS12] and [Pen12]), and applications to data structures and search trees (see [STT88] and [Deh10], for instance). ...
... Still, they fit well with the recently emerging connections between Thompson's groups, the geometry of mapping class groups of surfaces, and braid groups. See again [FKS12] and [Pen12] for relevant recent surveys of this emerging interaction. ...
... In other examples, the group G is the universal mapping class group of genus zero, or the asymptotically rigid mapping class group of infinite genus. See [FKS12], where these kinds of extensions are surveyed and put into a unifying context of cosimplicial symmetric group extensions. The authors also describe the action of the Grothendieck-Teichmüller group on some completions of the groups involved. ...
Article
We prove that Thompson's group V is acyclic. The strategy of our proof stems from the context of homological stability and stable homology. We first use algebraic K-theory methods to compute the stable homology for automorphism groups of algebraic theories in general, and for the Higman-Thompson groups in particular. This also leads us to the first examples where homological stability fails for automorphism groups of algebraic theories. We then provide a general homotopy decomposition technique for classifying spaces originating in McDuff's and Segal's work related to foliations to give a useable characterization of acyclicity. Finally, we apply this to deal with the specific features of Thompson's group.
... For our purposes, the main result we need about Thompson's group T is that its elements can be identified with pairs of ideal Farey m-gons in the dyadic Farey tessellation of the unit disc with an indication of how the corners of the one are sent to the corners of the other. This description induces piecewiselinear maps on the boundary circle (parametrized by arc length and rescaled dividing by 2π) [FKS12,Fos12]. Alternatively, its elements can be identified with the group of finite retriangulations and renormalizations of the Farey tessellation of the disc. ...
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Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degree 2 central extension of Thompson's infinite finitely presented simple group T with the cyclic group of order 2. These results are inspired by a similar recent analysis by the first author of the automorphism group of an infinite-dimensional associahedron.