"Thompson's Groups: New Developments and Interfaces"
Marseilles, CIRM, June 2 -6, 2008.
Abstracts of the talks
MINICOURSES
BRIN Matt:
Relatives of Thompson's groups
1. Thompson & McKenzie's finitely presented group with an unsolvable word problem
2. Braided groups
3. Higher dimensional groups
GHYS Etienne and SERGIESCU Vlad
Cohomology of Thompson's groups: old results!
MORRIS Dave
Using left-invariant orders to study actions on 1-manifolds
TALKS
BABENKO Ivan
On the group of power series with integer coefficients
We study certain properties of the groupe $J(Z)$ of substitutions of formal power series in one variable with integer coefficients. We show that $J(Z)$, regarded as a topological group, has four generators and cannot be generated by fewer elements. In particular, we show that the one-dimensional continuous homology of $J(Z)$ is isomorphic to $Z+Z+Z_2+Z_2$. We study various topological and geometric properties of the coset space $J(R)/J(Z)$> We compute the real cohomology $H^*(J(Z), R)$ with uniformly locally constant supports and show that it is naturally isomorphic to the cohomology of the nilpotent part of the Lie algebra of formal vector fields on the line.
BLEAK Colin
Algorithmic processes in Thompson's groups
Given a finite collection Gamma = {x_1, ..., x_n} of elements in $F$, we will discuss algorithms to detect 1) whether or not the group G generated by Gamma is solvable, and 2) in the case that G is solvable, a) its precise derived length, and b) whether a new element $y$ of $F$ can actually be found in $G$ (this last algorithm is not yet complete). This work is joint with John Meakin and Susan Hermiller.
Time allowing, we will also discuss some slightly older work joint with Kassabov and Matucci, which solves the simultaneous conjugacy problem in $T$.
BURILLO Jose
Metric properties of higher-dimensional Thompson's groups
In this talk we will describe some of the metric properties of the higher-dimensional Thompson's groups described by Brin. In these groups, the number of carets in the minimal binary tree diagram is no longer a lower bound for the metric, there are elements whose length is only log of the number of carets. We will observe that the exitence of carets of two types brings new types of distortion inside these groups. In particular, the groups F, T and V are exponentially distorted inside 2V.
CLEARY Sean
Random subgroups of Thompson's group F
(joint work with Murray Elder, Andrew Rechnitzer and Jennifer Taback)
There are a number of possible notions of constructing k-generator subgroups "at random" from a fixed group G. Given such a process, we can try to understand properties that a random subgroup has. For random subgroups of Thompson's group F, a number of interesting phenomena occur which are not present in other known examples. For example, there are positive densities of many isomorphism classes of k-generator subgroups, rather than there just being one isomorphism class of density 1. I will also describe a persistence phenomenon seen in Thompson's group, where some isomorphism classes of subgroups are present with positive density in the space of k-generator subgroups for all k larger than some K, with respect to one of the natural processes for constructing subgroups at random.
DEHORNOY Patrick
Dual presentation of Thompson's group $F$ and flip distance between triangulations
Viewing the group $F$ as the geometry group of associativity---as in Thompson's original approach---leads to expressing $F$ as the group of fractions of a monoid $F+*$ that is larger than the submonoid generated by the standard elements $x_i$. We show that $F+*$ admits left and right least common multiples and greatest common divisors. This gives a lattice structure on $F$, connected with the Tamari lattices. The associated metric is rather strange, as $F+*$ does not quasi-isometrically embed in $F$.
As an application, we consider the rotation distance problem, which is also the problem of finding the flip distance between triangulations of a polygon, and the diameter of the associahedra. The problem was asymptotically solved by Sleator, Tarjan, and Thurston using hyperbolic geometry. Using combinatorial arguments derived from the above approach, we prove lower bound results that are non-optimal, but nevertheless non-trivial.
FARLEY Dan
Actions of Brin's Groups nV on CAT(0) cubical complexes.
Suppose that the group G acts on a compact ultrametric space. Bruce Hughes has shown that if the action of G is locally determined by a finite similarity structure, then G acts properly on a CAT(0) cubical complex and is (therefore) a-T-menable. Brin introduced a collection of groups nV, which are groupsof homeomorphisms of the n-fold product of the Cantor set. In this talk, I will describe joint work with Bruce Hughes in which we showthat Brin's groups nV act properly by isometries on CAT(0) cubical complexes and are a-T-menable.
The main idea is to extend Hughes's earlier results in order to handle actions on products of compact ultrametric spaces.
FUNAR Louis
The braided Ptolemy-Thompson group T and applications to the quantization of Teichmuller spaces
The braided Thompson group T^* is an extension of the Ptolemy-Thompson group T by the infinite braid group B_infty. Using their interpretation as mapping class groups of infinite surfaces we prove that T and T^* are (asynchronously) combable (joint work with C.Kapoudjian).
The quantization of the Teichmuller space (as defined by Chekhov, Fock, Goncharov and Kashaev) yields unitary projective representations of mapping class groups and in particular of the group T. We prove that the corresponding central extension of T is 12 times the Euler class (joint work with V.Sergiescu).
GEOGHEGAN Ross
New invariants of Thompson's group $F$
We calculate the Bieri-Neumann-Strebel-Renz Sigma-invariants for Thompson's group $F$ (joint with Bieri and Kochloukova). We also prove that the Product Conjecture for Sigma-invariants is true for the product $F x F$ (even though it is now known to be false in general for products $G x H$). From these results we deduce: For each $n geq 0$ there is a subgroup of $F$ of type $FP_n$ which is not of type $FP_{n+1}$.
GEOGHEGAN Ross
The Whitehead group of T
This is a preliminary report on joint work with Marco Varisco. We apply some new work of the Muenster school, specifically of Lueck-Reich-Rognes-Varisco, to show that Wh(T), the Whitehead group of Thompson's group T, is non-trivial. In fact Wh(T) is large: its tensor product with the rationals is an infinite-dimensional rational vector space.
GONZALEZ-MENESES Juan
Bi-orderings on pure braided Thompson's groups
(joint work with Jose Burillo)
Recently and independently, Brin and Dehornoy introduced braided versions of Thompson's groups, in which permutations are replaced by braids. In this context, the pure braided Thompson's group BF is a subgroup of the braided Thompson's group BV, in the same way as the group of pure braids is a subgroup of the braid group $B_infty$. We show a property that parallels the corresponding one for braids and pure braids: the pure braided Thompson's group BF is biorderable, while BV is orderable but not bi-orderable.
HUGHES Bruce
Local similarities and the Haagerup property
A new class of groups, the locally finitely determined groups of local similarities on compact ultrametric spaces, is introduced and it is proved that these groups have the Haagerup property (that is, they are a-T-menable in the sense of Gromov). The class includes Thompson's groups, which have already been shown to have the Haagerup property by D. S. Farley, as well as many other groups acting on boundaries of trees.
KAPOUDJIAN Christophe
Extensions of Thompson's group V
We present an algebraic formalism which enables us to describe in a unifying way several recently discovered extensions of Thompson's group V, among which the braided Thompson group of Brin-Dehornoy and the universal mapping class group of Funar and myself. It is based on the existence of ``strand doubling maps" $partial^n_i:S_n rightarrow S_{n+1}$ relating the symmetric groups. This formalism is quite useful to explain why and how the Grothendieck-Teichmueller group acts on some completions of these groups.
MATUCCI Francesco
Structure theorems for subgroups of homeomorphisms groups
(joint work with C. Bleak and M. Kassabov).
We give a classification of the solvable subgroups G of the group Homeo_+(S^1) of all orientation-preserving homeomorphisms of the unit circle. The key tool is proving that the rotation number map is a group homomorphism and it is done by relating the dynamics of G and its group structure. Applications include new proofs of known results as the Margulis' theorem on the existence of a G-invariant probability measure on S^1 and refining a theorem of Ghys on the classification of solvable groups of analytic diffeomorphisms.
MURANOV Alexei
Arithmetic in groups of piecewise affine permutations of an interval
(joint work with Tuna Altinel)
Bardakov and Tolstykh recently showed that the Richard Thompson's group $F$ interprets the Arithmetic $(Z,+,x)$ with parameters. We consider a class of infinite groups of piecewise affine permutations of an interval (groups of Bieri and Strebel) which contains all the three groups of Thompson and some classical families of finitely presented infinite simple groups. We interpret Arithmetic in all groups of this class. In particular we show that the elementary theories of all these groups are undecidable. Furthermore, we interpret Arithmetic in $F$ and some of its generalisations without parameters.
ROEVER Claas
Commensurations and quasi-isometries of Thompson's group F
I will present a complete description of the abstract commensurator C of Thompson's group F. One consequence is that every finite-index subgroup of F is virtually F and that there exist such subgroups which are not isomorphic to F. I will then, to some extend, describe the algebraic structure of C. Finally, I will explain how this leads to the first non-trivial examples of quasi-isometries of F.
THOMPSON Richard
The semigroup context
The group operating on parenthesized expressions (which amounts to V) is generated -- even in the semigroup sense -- by the elements mapping ab to ba, a(bc) to (ab)c, a(bc) to a(cb), and a(b(cd)) to a((bc)d). If we add the operations mapping a to aa (call this U) and ab to b (call this L) the first two operations become definable from the others and we obtain a semigroup S which is anti-isomorphic to the semigroup of finite binary table transformations (where we weaken the permutation case requirement for right-hand sequences to merely prohibiting partial overlaps). Thus L corresponds to the table operation L' given by 0 mapsto 10 and 1 mapsto 11, and U to the table operation U' give n by 00 mapsto 0, 01mapsto 1, 10 mapsto 0, and 11 mapsto 1. (So UL corresponds to L'U', or the identity.) There is a nice conceptual proof of a finite presentation of S. And S has the property that for all v, w, x, and y in S with v not equal to w there exist p and q in S with both pvq = x and pwq = y, so adding a new relation to S collapses it quickly. The semigroup S is also useful in finitizing logic and dealing with automata on trees.
WLADYS Claire
Metric behavior of generalizations of Thompson's group F
There exist generalizations of Thompson's group F (also written as F(2)) which depend upon more than one integer and are of the form F(n_1,...,n_k) for integers k and n_i greater than 1. Using tree-pair diagram representatives, we will describe several metric properties of a class of these generalized Thompson's groups which reveal behavior which is distinct from the metric behavior of F. The uniqueness of minimal tree-pair diagram representatives, the relationship of the metric to the number of leaves in minimal tree-pair diagram representatives, and the ability to quasi-isometrically embed F and other Thompson groups into these generalized groups are all distinctly different than in F or in F(n) (for any integer n>1).
ZUK Andrzej
Amenability
The question about amenability of the Thompson group had a great impact on the study of the asymptotic properties of discrete groups. In recent years fundamental problems concerning amenable groups were solved. We would like to present latest developments in this subject related to growth (intermediate, non-uniform, ...) for automata groups.