The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V

Divergence properties of the generalised Thompson groups and the braided-Thompson groups

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.

On normal subgroups of the braided Thompson groups

We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such normal subgroups. For Fbr, we prove that for any normal subgroup N of Fbr, either N is contained in the kernel of Fbr \onto F, or else N contains [Fbr,Fbr]. We also compute the Bieri-Neumann-Strebel invariant Sigma^1(Fbr), which is a useful tool for understanding normal subgroups containing the commutator subgroup.

Bi-orderings on pure braided Thompson's groups

In this paper it is proved that the pure braided Thompson's group BF admits a bi-order, analog to the bi-order of the pure braid groups.

辫子Thompson群的散度函数

Divergence function of the braided Thompson group

本文证明了辫子Thompson群$BV$具有线性散度函数，并据此得出其所有渐近锥均无割点的结论。

由 Ore 范畴产生的可序 Thompson 型群

Orderable Thompson-like groups arising from Ore categories

本文通过间接因子建立 Ore 范畴基本群左可序与双可序的充分条件，并证明数字重写系统的辫分式群左可序、纯辫分对应群双可序。

A new family of infinitely braided Thompson's groups

We present a generalization of the Dehornoy-Brin braided Thompson group $BV_2$ that uses recursive braids. Our new groups are denoted by $BV_{n,r}(H)$, for all $n\geq 2,r\geq 1$ and $H \leq \mathcal{B}_n$, where $\mathcal{B}_n$ is the braid group on $n$ strands. We give a new approach to deal with braided Thompson groups by using strand diagrams. We show that $BV_{n,r}(H)$ is finitely generated if $H$ is finitely generated.

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.

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.

The braided Thompson's groups are of type $F_\infty$

We prove that the braided Thompson's groups $V_{\rm br}$ and $F_{\rm br}$ are of type $F_\infty$, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected. In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups of the pure braid group that are highly generating, in the sense of Abels and Holz.

Embeddings into Thompson's groups from quasi-median geometry

The main result of this article is that any braided (resp. annular, planar) diagram group $D$ splits as a short exact sequence $1 \to R \to D \to S \to 1$ where $R$ is a subgroup of some right-angled Artin group and $S$ a subgroup of Thompson's group $V$ (resp. $T$, $F$). As an application, we show that several braided diagram groups embeds into Thompson's group $V$, including Higman's groups $V_{n,r}$, groups of quasi-automorphisms $QV_{n,r,p}$, and generalised Houghton's groups $H_{n,p}$.

Geometric structures related to the braided Thompson groups

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $Σ^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $Σ^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is f…

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。

Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups

This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^\sharp,T^\ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T…

Conjugacy and Dynamics in Thompson's Groups

We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.

The Algebra of Strand Splitting. II. A Presentation for the Braid Group on One Strand

Presentations are computed for a braided version BV of Thompson's group V and for V itself showing that there is an Artin group/Coxeter group relation between them. The presentation for V is obtained from that for BV by declaring all that all generators are involutions.

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。

Thompson's Group F

We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for…

Braided Diagram Groups and Local Similarity Groups

Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common characteristics: both act properly by isometries on CAT(0) cubical complexes, and certain groups in both classes have type F-infinity, for instance. Here we clarify the relationship between these families of groups: the braided diagram groups over tree-like semigroup presentations are precisely the groups that act on compact ultrametric spaces via small similarity str…

辫状 Higman–Thompson 群的亲属群的有限性性质

Finiteness properties for relatives of braided Higman--Thompson groups

本文刻画带辫群标签的 bV、bT 与 bF 群的 F_n 型有限性，证明其与标签群 H 等价，并证实 Aroca–Cumplido 猜想。

Asymptotically rigid mapping class groups and Thompson's groups

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 Teichmüller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results.

Dynamics in Thompson's Group F

We describe an explicit relationship between strand diagrams and piecewise-linear functions for elements of Thompson's group F. Using this correspondence, we investigate the dynamics of elements of F, and we show that conjugacy of one-bump functions can be described by a Mather-type invariant.

类Thompson群，Reidemeister数与不动点

Thompson-like groups, Reidemeister numbers, and fixed points

本文研究了类Thompson群$F$的自同构不动点性质，推导出一种上同调判据来检测阿贝尔化中的无限不动点集并蕴含性质$R_\infty$，利用BNS $Σ$-不变量证明该工具广泛适用于多种$F$类群。

不可约边移位的拓扑全群具有可解共轭问题

Topological Full Groups of Irreducible Edge Shifts have Solvable Conjugacy Problem

本研究利用链图和交换环半群等方法，解决了不可约边移位的拓扑全群的共轭问题，相关技术可推广至包含Houghton群、类Thompson群$QV$的更大边移位分段典范同胚群类。

Von Neumann algebras of Thompson-like groups from cloning systems

We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under …

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.

Finiteness properties of some groups of local similarities

Hughes has defined a class of groups, which we call FSS (finite similarity structure) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson's group V. Guided by previous work on Thompson's group V, we establish a number of new results about FSS groups. Our main result is that a class of FSS groups are of type F-infinity. This generalizes work of Ken Brown from the 1980s. Next, we develop methods for distinguishing between isomorphism types of some of the Nekrashevych-Röver groups V_{d}(H), where H is a finite group, and show that a…

On Belk's classifying space for Thompson's group F

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group PB_n. Elements of Thompson's group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk's thesis, a space CF was considered, of configurations of points on the real line allowing for splitting and merging, and a proof was sketched that CF is a classifying space for F. The idea there was to build the universal cover and construct an explicit contraction to a point. Here we start with an established CAT(0) cube complex X on which F acts…

Operad groups and their finiteness properties

We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.

The group of parenthesized braids

We investigate a group $B\_\bullet$ that includes Artin's braid group $B\_\infty$ and Thompson's group $F$. The elements of $B\_\bullet$ are represented by braids diagrams in which the distances between the strands are not uniform and, besides the usual crossing generators, new rescaling operators shrink or strech the distances between the strands. We prove that $B\_\bullet$ is a group of fractions, that it is orderable, admits a non-trivial self-distributive structure, i.e., one involving the law $x(yz)=(xy)(xz)$, embeds in the mapping class group of a sphere with a Cantor set of punctures, a…

Classifying spaces from Ore categories with Garside families

We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The Zappa--Szép product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our m…

分形重排群中的共轭性

Conjugacy in Rearrangement Groups of Fractals

本文推广链图方法，利用两类重写系统解决多种分形重排群及相关 Thompson 型群的共轭问题，并为 Houghton 群和 Higman–Thompson 群给出新解法。

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Quantization of universal Teichmüller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite bra…