An extension of the Burau representation to a mapping class group associated to Thompson's group T

The braided Ptolemy-Thompson group $T^*$ is asynchronously combable

The braided Ptolemy-Thompson group $T^*$ is an extension of the Thompson group $T$ by the full braid group $B_{\infty}$ on infinitely many strands. This group is a simplified version of the acyclic extension considered by Greenberg and Sergiescu, and can be viewed as a mapping class group of a certain infinite planar surface. In a previous paper we showed that $T^*$ is finitely presented. Our main result here is that $T^*$ (and $T$) is asynchronously combable. The method of proof is inspired by Lee Mosher's proof of automaticity of mapping class groups.

Remarks on the faithfulness of the Jones representations

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4.

The braided Ptolemy-Thompson group is finitely presented

Pursueing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group $T^*$ (and its further generalizations) which is an extension of the Ptolemy-Thompson group $T$ by means of the full braid group $B_{\infty}$ on infinitely many strands. We prove that it is a finitely presented group with solvable word problem, and give an explicit presentation of it.

带与不带拟同态的辫化汤普森群

Braided Thompson groups with and without quasimorphisms

本研究分析了多种辫化汤普森群的拟同态与有界上同调，证明$bV$等群有无穷维拟同态空间，而$\bV$二阶有界上同调平凡，提供了相关特殊群类的首个实例。

Diffeomorphisms groups of tame Cantor sets and Thompson-like groups

The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise when we consider products of central ternary Cantor sets. We derive that the $\mathcal C^2$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Point pushing, homology, and the hyperelliptic involution

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As a consequence, we show that the hyperelliptic Torelli group is generated by Dehn twists if and only if it is generated by reducible elements. We also give an application to the kernel of the Burau representation.

Centrally extended mapping class groups from quantum Teichmuller theory

The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmüller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in \cite{FS} for the Thompson groups.

On the automorphism group of the asymptotic pants complex of a planar surface of infinite type

We consider a planar surface Σof infinite type which has the Thompson group T as asymptotic mapping class group. We construct the asymptotic pants complex C of Σand prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the Thompson group T by Z/2Z.

渐近刚性映射类群 II：链图与非正曲率

Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature

该研究提出由特定半群表示的编织链图构造的Chambord群新类，涵盖多种已知渐近刚性映射类群，并证明其多循环子群具有几乎阿贝尔且不失真的性质。

On the automorphisms group of the asymptotic pants complex of an infinite surface of genus zero

The braided Thompson group $\mathcal B$ is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher-Thurston cell complex whose vertices are asymptotically trivial pants decompositions. We prove that the automorphism group $\hat{\mathcal B^{\frac{1}{2}}}$ of this complex is also an asymptotic mapping class group in a weaker sense. Moreover $\hat{\mathcal B^{\frac{1}{2}}}$ is obtained by $\mathcal B$ by first adding new elements called half-twists and further comple…

On a universal mapping class group of genus zero

The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a {\it universal mapping class group of genus zero}, denoted $\B$. The latter is a nontrivial extension of the Thompson group $V$ (acting on the Cantor set) by an inductive limit of pure mapping class groups of all genus zero surfaces. We prove that $\B$ is a finitely presented group, and give an explicit presentation of it.

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal functor from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of "pointed Lagrangian relations" between skew-Hermitian $R$-modules. We call it the "Magnus functor" since it contains the Magnus representation of mapping class groups as a s…

通用旋量泰希米勒理论，II. P(SL(2,Z))的有限表示

Universal Spin Teichmueller Theory, II. Finite Presentation of P(SL(2,Z))

该研究定义了旋量映射类群P(SL(2,Z))，通过计算旋量覆叠的轨形甲板群得到其有限表示，还发现对易关系与根格的关联并推测其自同构群为散在群，配套论文验证了群的拓扑对应性。

块映射类群及其有限性性质

Block mapping class groups and their finiteness properties

本文定义了由子群H规定局部作用的块映射类群，证明当块为球面或环面时其有限性性质与H一致，并推证了Cantor曲面映射类群中特定有限性子群的存在性。

Representation stability in the level 4 braid group

We investigate the cohomology of the level 4 subgroup of the braid group, namely, the kernel of the mod 4 reduction of the Burau representation at $t=-1$. This group is also equal to the kernel of the mod 2 abelianization of the pure braid group. We give an exact formula for the first Betti number; it is a quartic polynomial in the number of strands. We also show that, like the pure braid group, the first homology satisfies uniform representation stability in the sense of Church and Farb. Unlike the pure braid group, the group of symmetries - the quotient of the braid group by the level 4 subg…

Sketch of a Program for Universal Automorphic Functions to Capture Monstrous Moonshine

We review and reformulate old and prove new results about the triad $ {\rm PPSL}_2({\mathbb Z})\subseteq{\rm PPSL}_2({\mathbb R})\circlearrowright ppsl_2({\mathbb R}) $, which provides a universal generalization of the classical automorphic triad ${\rm PSL}_2({\mathbb Z})\subseteq{\rm PSL}_2({\mathbb R})\circlearrowright psl_2({\mathbb R})$. The leading P or $p$ in the universal setting stands for $piecewise$, and the group ${\rm PPSL}_2({\mathbb Z})$ plays at once the role of universal modular group, universal mapping class group, Thompson group $T$ and Ptolemy group. We produce a new basis o…