arXiv research

New family of braided Thompson groups introduced using recursive braids.

Study of quasimorphisms and bounded cohomology in braided Thompson groups.

New virtual version of Thompson's group created to handle virtual knots.

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.

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 …

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…

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 …

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 s…

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 …

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 study proves properties of specific groups acting on cube complexes.

The central extension of the Thompson group $T$ that arises in the quantized Teichmüller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extension…

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…

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 th…

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 …

New groups from strand diagrams show polycyclic subgroups are virtually abelian and undistorted.

We study some aspects of the geometric representation theory of the Thompson and Neretin groups, suggested by their analogies with the diffeomorphism groups of the circle. We prove that the Burau representation of the Artin braid groups extends to a mapping class group $A_T$ related to Thompson's group $T$ by a short e…

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 ext…

In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an orient…

Introduces halo products and studies their geometric properties.

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 …

Positive Thompson links are arborescent tangles.

Extends Jones' construction to Thompson's group F and link homology.

Jones constructs knots from Thompson group elements.

Positive Thompson links are proven for oriented subgroup elements.

Brin-Thompson groups have new properties for n>=2.

New $p$-colorable subgroup derived from Thompson's group.

New method counts link components from Thompson group elements.

New method associates annular links to elements of Thompson's group T.

New methods use Conway tangles to generate knots and links.

New groups can't be fundamental groups of symplectic Calabi-Yau manifolds.

Paper proves almost all stabilizer subgroups of Thompson's group satisfy Alexander's theorem.

This paper explores links from Thompson's group conjugacy classes.

We introduce a groupoid ${\mathbf{ΠMG}}}$, called the fundamental modular groupoid, which is a variant of Penner's mapping class groupoid. We study how it relates to the surface mapping class groups and Thompson's group $\mathsf T$. We also introduce larger groupoid $\mathbf{ΩMG}$, which is related to outer automorphis…

Half grid diagrams prove every link can be represented by a special type of grid diagram.

We review recent developments in the theory of Thompson group representations related to knot theory.

Classic braids embed in virtual braids.

We prove that the Brin-Thompson groups sV, also called higher dimensional Thompson's groups, are of type F_\infty for all natural numbers s. This result was previously shown for s up to 3, by considering the action of sV on a naturally associated space. Our key step is to retract this space to a subspace sX which is ea…

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…

In this note we prove that Thompson's group F cannot be the fundamental group of a symplectic 4-manifold with trivial canonical class by showing that its Hausmann-Weinberger invariant q(F) is strictly positive.

The paper examines subgroup separability for surface and virtual braid groups.

Classifies isotopy classes of links from Thompson's group F and its subgroup.

The paper finds minimal generating sets and abelianizes the quasitoric braid group.

Study virtual braid groups, proving a key subgroup result.

We show how to construct unitary representations of the oriented Thompson group $\vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec{F}$.

We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite loop space of the mod $n-1$ Moore spectrum. As $V = V_{2,1}$, we can deduce that t…

Paper explores relations between braid groups and their quotients.

Paper proves homotopy braid group properties over integers and three strands.