Narrow Results

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 prove a computable version of Hall’s Harem theorem and apply it to computable versions of Tarski’s alternative theorem.

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.

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In Part I, we prove that a disjoint union has property A of Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).