
Abstracts
Workshop on the Thompson Groups
Odense, August 1517, 2016.
Title: Rearrangement Groups of Fractals.
Abstract:
We will describe a family of Thompsonlike groups that act on selfsimilar spaces. The spaces that these groups act upon can be constructed explicitly as limits of systems of finite graphs, with elements of the group corresponding to isomorphisms between the graphs. Each such group acts on an associated CAT(0) cubical complex as well, and by analyzing the geometry of this complex we prove that some of the groups have type F infinity. This is joint work with Bradley Forrest.
Title: On subgroups of Thompson's group F in which F does not embed.
Abstract:
We have been able to analyze a family of groups described in
the title of greater complexity than previously known. While
algebraically complex, the family is surprisingly easy to describe and
surprisingly cooperative when calculating the degree of complexity.
Title: Normal Subgroups of the LodhaMoore Groups.
Abstract:
In 2013, Lodha and Moore described two finitely presented groups which contain Thompson's group F, and which are counterexamples to the Von Neumann conjecture, i.e., they are not amenable and they do not contain nonabelian free subgroups. In this talk I will give a small introduction to these groups and I will describe some of their properties, showing how one can work with them with treepair diagrams. Then I will show joint results with Yash Lodha and Lawrence Reeves, which show that these groups, as well as the Monod groups, have simple (first or second) commutator. In particular, if L is the small LodhaMoore group, then its commutator L' is simple, in a similar fashion to F. We will speculate on whether this gives any indication to the amenability of F.
Title: Using random walks to compute cogrowth and decide amenability for finitely generated groups.
Abstract:
I will report on my work with Rechnitzer and van Rensburg, and new work with my PhD student Cameron Rogers, on algorithms that use random walks to experimentally compute cogrowth rates and functionvalues for arbitrary finitely presented groups. In particular we apply the algorithms to Thompson's group F and get some interesting results.
Title: Thompson's group F is probably not amenable.
Abstract:
We have computed a further 7 moments of the operator (A+B+A^(1)+B^(1)) in Thompson's group F, bringing the total to 31 known coefficients. According to our analysis of this series, these moments appear to grow at the rate mu^n kappa^(n^(sigma)), multiplied by less dominant terms, where mu≈14.9, sigma≈1/2 and kappa<1. For F to be amenable we would have to have mu=16. Moreover, we prove that if F is amenable then the subdominant term of the growth rate cannot have the form kappa^(n^(sigma)) for any kappa and sigma. This is joint work with Tony Guttmann and builds on recent work by Søren Haagerup, Uffe Haagerup and Maria RamirezSolano.
Title: The generation problem in Thompson group F.
Abstract:
We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin.
Title: Densities of Cayley graphs of F in various generators.
Abstract:
Let G be a group generated by a set A of cardinality m. For all nonempty finite subgraphs of its Cayley graph Gamma, we take the least upper bound of the average vertex degree. This is called the density of Gamma.
It is well known that G is amenable if and only if its density equals 2m. So one can ask what is known about F in this direction. In the
standard generating set x_0,x_1, there is an example constructed by Belk and Brown. It shows that there are finite subgraphs in the
Cayley graph with average vertex degree approaching 3.5. So the density is at least 3.5. At the present moment, nobody could exceed
this estimate.
The strong version of nonamenability conjecture states that the above example is optimal. We study equivalent versions of the
same conjecture in different generating sets. For instance, in the symmetric generating set x_1, x_1x_0^(1). Notice that F has an
automorphism sending x_0 to x_0^(1) and the above generators to each other. The Cayley graph in these two generators has many
interesting properties and it deserves a separate study (say, in terms of the growth function). For this graph, the above example
by Belk and Brown, gives the estimate for the density of the Cayley graph only 3. This is equivalent to the strong amenability conjecture.
Also we look at the generating set x_0,x_1,x_2 and study its properties in the sense of density. Notice that F has many properties of
selfsimilarity, so this approach can help for the (non)amenability problem in general.
Title: Some unitary representations of the Thompson groups.
Abstract:
The Thompson groups F, T, and V can be realised as "groups of fractions" of certain
categories of forests. An analysis of the group of fractions construction shows that any functor from the category in question
to another yields an action of the group of fractions on a direct limit space. For these categories
of forests, functors may easily be constructed from standard data in subfactors/TQFT. The coefficients of the
unitary representations can be interesting and in particular give a geometric way to construct all knots and links
from F.
Title: The Thompson groups and the Cuntz algebra O_2.
Abstract:
As shown by V. Nekrashevych [2004], the Thompson group V can be realized naturally as a subgroup of the unitary group of the Cuntzalgebra O_2. We study the unital endomorphisms of O_2 which preserve the Thompson group F using “snake” diagrams. The main focus will be on a certain subsemigroup of endomorphisms preserving F, which we show to be in bijection with the palindromes of base 3 of certain lengths. This is joint work with Selcuk Barlak.
In the second part of my talk, I will discuss how we can estimate the norm of a certain element in the complex group ring of T, considered as an operator via the left regular representation of T. By a result of U. Haagerup and K. Olesen, it follows that the value of this norm is closely related to the amenability problem of the Thompson group F. This is joint work with Uffe Haagerup and Søren Haagerup, which is a continuation of the work started in “A computational approach to the Thompson group F”, by the same authors.
Title: Approximate enumeration of trivial words.
Abstract:
In recent work with Elder and van Rensburg, we introduced a random sampling algorithm for trivial words in finitely presented groups based on an statistical physics algorithm for sampling simple loops embedded in regular graphs.
One drawback of that algorithm was that it had difficulty sampling long trivial words. In this talk I will describe how this algorithm can be extended to ameliorate this problem using "flathistogram" ideas. As a pleasant sideeffect, the new algorithm also estimates the number of trivial words
Title: Implications of quickly growing Folner functions for cogrowth.
Abstract:
Many characterisations of amenability are defined in terms of limits. Connections between the rates of convergence of these limits can compromise numerical algorithms for determining amenability. In this talk the connections between cogrowth and the Folner function are explored.
Title: On subgroups of the R. Thompson group F.
Abstract:
We study stabilizers H_U of subsets U of [0,1] in F. In particular, we describe the structure of H_U if U is finite. We show that if U is finite, then H_U is finitely generated if and only if U does not contain irrational numbers. If finite U, V consist of only rational numbers which are not finite dyadic fractions, then H_U is isomorphic to H_V if and only if U=V. In that case H_U and H_V are conjugate inside a completion of F with respect to a natural "Hamming" metric which is a subgroup of Homeo([0,1]). This is a joint work with Gili Golan.

