Narrow Results

We study some properties of the Cayley graph of R. Thompson's group F in generators x₀, x₁. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x₀, x₁. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x₀, x₁ and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of .

Let F denote the Thompson group with standard generators A = x₀, B = x₁. It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent to

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z}$, the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Brin–Navas group $B$. We have also given an improved algorithm for the coefficients of Thompson’s group $F$, giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson’s group F using the method from [S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp.25 (2015) 381–432] applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability.

Again for Thompson’s group F we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability.

We estimate the asymptotics of the cogrowth coefficients of F to be

We show every locally solvable subgroup of $\mathrm{\text{PLo}}(I)$ is countable. A corollary is that an uncountable wreath product of copies of $\mathbb{Z}$ with itself does not embed into $\mathrm{\text{PLo}}(I)$.

By the density of a finite graph we mean its average vertex degree. For an $m$-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with $m$ generators is amenable if and only if the density of the corresponding Cayley graph equals $2m$.

A famous problem on the amenability of R. Thompson’s group $F$ is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators $\{x_0,x_1\}$, is at least $3.5$. This estimate has not been exceeded so far.

For the set of symmetric generators $S=\{x_1,{\bar{x}}_1\}$, where ${\bar{x}}_1=x_1{x}_0^{-1}$, the same example only gave an estimate of $3$. There was a conjecture that for this generating set equality holds. If so, $F$ would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set $X\subset F$, the inequality $\left|S^{\pm 1}X\right|\ge 2\left|X\right|$ holds.

In this paper, we disprove this conjecture showing that the density of the Cayley graph of $F$ in symmetric generators $S$ strictly exceeds $3$. Moreover, we show that even larger generating set $S_0=\{x_0,x_1,{\bar{x}}_1\}$ does not have doubling property.

We study subgroups of the group $\mathrm{\text{PLo}}(I)$ of piecewise linear orientation-preserving homeomorphisms of the unit interval $I=[0,1]$ that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of $\mathrm{\text{PLo}}(I)$ which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application, we give a necessary and sufficient condition for any two subgroups of the R. Thompson group $F$ that are stabilizers of finite sets of numbers in the interval $(0,1)$ to be isomorphic, thus solving a problem by G. Golan and M. Sapir. We also show that if two stabilizers are isomorphic, then they are conjugate inside a certain group ${𝒢}$.

In this paper, we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that the existence of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These schemes are called pure. We obtain a criterion for the existence of such a scheme in terms of isoperimetric constant of the graph. We analyze R. Thompson’s group $F$, for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators $\{x_0,x_1,{\bar{x}}_1\}$, where ${\bar{x}}_1=x_1{x}_0^{-1}$. However, the question becomes open if edges with labels $x_0^{\pm 1}$ can be used twice. The existence of pure evacuation schemes for this version is implied by some natural conjectures.

Amenability of some discrete subgroups of the group of diffeomorphisms of interval is proved. As a consequence, a solution of the problem of amenability of the Thompson group F is given.