Related papers: From skein theory to presentations for Thompson gr…

We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don't cross, until you eventually get a linear combination of links with no…

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…

We consider the infinite symmetric group and its infinite index subgroup given as the stabilizer subgroup of one element under the natural action on a countable set. This inclusion of discrete groups induces a hyperfinite subfactor for each…

We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

We construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq…

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…

There is a map, defined and studied by Jones, from Thompson's group $F$ to knots. Jones proved that every knot is in the image of this map -- that is, that every knot can be seen as the "knot closure" of a Thompson group element. We…

In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups have remarkable diagrammatic properties that can be used to…

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…

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…

In this paper we show that the cloning system construction of Skipper and Zaremsky [SZ21], under sufficient conditions, gives rise to Thompson-Like groups which are stable; in particular, these are McDuff groups in the sense of Deprez and…

The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. This is a long standing open problem which has attracted a lot…

This paper demonstrates the uniformly finite homology developed by Block and Weinberger and its relationship to amenable spaces via applications to the Cayley graph of Thompson's Group F. In particular, a certain class of subgraph of F is…

Cayley's theorem tells us that all groups $\mathbf{G}$ occur as subgroups of the group of automorphisms over some set $X$. In this paper we consider a `sort-of' converse to this question: given a set $X$ and some transformation group…

We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a…

Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup J of GL(3,Z12) generated by the three voicing reflections. As applications of our Structure Theorem, we determine the structure of the…

Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest…

We initiate the systematic study of modular representations of symmetric groups that arise via the braiding in (symmetric) tensor categories over fields of positive characteristic. We determine what representations appear for certain…

We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the two-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum parameter. We obtain formulas for the dimension of the skein module of $T^3$, and we describe…