Tag - Thompson groups

Susan Hermiller: Subgroups of the group of dyadic piecewise linear homeomorphisms of the real line

3rd May, 2024

The group of dyadic orientation-preserving piecewise linear (PL) homeomorphisms of the unit interval is called Thompson's group F, and the question of which groups are - or cannot be - subgroups of F has yielded many interesting results. In this talk I'll discuss the question of what groups can or cannot be subgroups of Aut(F) (the automorphism group of F), and more particularly subgroups of an index 2 subgroup of Aut(F) that is isomorphic to a group of dyadic PL homeomorphisms of the real line.

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Mark Sapir: Subgroups of the R.Thompson group

29th April, 2022

This is joint work with Gili Golan-Polak. We describe the so-called closed subgroups of F. In particular, we construct a subgroup of F with easily decidable membership problem and undecidable conjugacy problem.

Matt Brin: Ordinals and subgroups of Thompson’s group F

18th March, 2022

Subgroups of Thompson's group F can be very complex. We give a family of elementary amenable subgroups that models a large initial segment of countable ordinals. The family models not only the order structure but also the basic operations of sum, product and exponentiation with base ω. Part of the appeal of the family is its ease of description.

Matthew Zaremsky: Houghton-like groups from “shift-similar” groups

25th February, 2022

In joint work with Brendan Mallery, we introduce and study so called "shift-similar" groups. Self-similar groups are a well known class of groups, which in particular interact nicely with Higman-Thompson groups, and we introduce shift-similar groups as an analog that interacts nicely with Houghton groups. Shift-similar groups actually turn out to have many properties that self-similar groups do not, for example every finitely generated group embeds into some finitely generated shift-similar group, and there exist uncountably many finitely generated shift-similar groups. In this talk I will recall some background on self-similar groups, introduce shift-similar groups and the Houghton-like groups they produce, and discuss the aforementioned results plus some results about amenability. I will also highlight some open questions.

Benjamin Steinberg: Cartan pairs of algebras

1st October, 2021

In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting "nicely" inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence).

In recent years, Matsumoto and Matui exploited this to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type using graph C*-algebras and their commutative subalgebras of continuous functions on the shift space (which form a Cartan pair under mild assumptions on the graph). The key point was translating these dynamical conditions into groupoid language. The ring theoretic analogue of graph C*-algebras are Leavitt path algebras. Leavitt path algebras are also connected to Thompson's group V and some related simple groups considered by Matui and others. Since the Leavitt path algebra associated to a graph is the "Steinberg" algebra of the same groupoid (a ring theoretic version of groupoid C*-algebras whose study was initiated by the speaker), this led people to wonder whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the shift space. The answer is yes, and it turns out in the algebraic setting one doesn’t even need any conditions on the graph. Initially work was focused on recovering a groupoid from the pair consisting of its "Steinberg" algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture.

It turns out that a twist on a groupoid gives rise to a Cartan pair when the algebra satisfies a groupoid analogue of the Kaplansky unit conjecture. In particular, if the groupoid has a dense set of objects whose isotropy groups satisfy the Kaplansky unit conjecture (e.g., are unique product property groups or left orderable), then the groupoid gives rise to a Cartan pair. This is what happens in the case of Leavitt path algebras where the isotropy groups are either trivial or infinite cyclic and hence left orderable.

Rachel Skipper: Maximal Subgroups of Thompson’s group V

16th October, 2020

There has been a long interest in embedding and non-embedding results for groups in the Thompson family. One way to get at results of this form is to classify maximal subgroups. In this talk, we will define certain labellings of binary trees and use them to produce a large family of new maximal subgroups of Thompson's group V. We also relate them to a conjecture about Thompson's group T.

Tony Guttmann: On the amenability of Thompson’s Group F

7th August, 2020

In 1967 Richard Thompson introduced the group F, hoping that it was non-amenable, since then it would disprove the von Neumann conjecture. Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including mention of the many incorrect "proofs" of amenability or non-amenability. Then I will give details of a new, efficient algorithm for obtaining terms of the co-growth sequence. Finally I will describe a number of numerical methods to analyse the co-growth sequences of a number of infinite, finitely-generated groups, and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an alternative route to a rigorous proof.

Collin Bleak: On the complexity of elementary amenable subgroups of R. Thompson’s group F

7th August, 2020

The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group G is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group F which is strictly well-ordered by the embeddability relation in type ε0 + 1. All except the maximum element of this family (which is F itself) are elementary amenable groups. In this way, for each α less than ε0, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is α + 2. The talk will is pitched for an algebraically inclined audience, but little background knowledge will be assumed.