On the properties of the Cayley graph of R. Thompson's group F








V. Guba
Vologda State Pedagogical University, Russia






Let $G$ be a group with $m$ generators and let $C$ be the corresponding Cayley graph. For any finite subgraph $\Gamma$ in $C$, let $d(\Gamma)$ be the average vertex degree of $\Gamma$. By $\bar d(C)$ we denote the least upper bound of all the $d(\Gamma)$, where $\Gamma$ runs over all finite subgraphs in $C$.

It is well-known that $G$ is amenable if and only if $\bar d(C)=2m$. There exists one more criterion of amenabilty. Namely, $G$ is non-amenable if and only if there exists a function $f\colon G\to G$ such that 1) the distance between $g$ and $f(g)$ in $C$ is bounded by a constant $K$ for all $g\in G$ and 2) each element in $G$ has at least two preimages under $f$.

Suppose that $K=1$ in the above definition. This means that for any $g\in G$, the element $f(g)$ is either $g$, or equals $ga$, where $a$ is a generator or its inverse. We say that the graph $C$ has a doubling property whenever the function $f$ with the above conditions exists for $K=1$.

The group $F$ can be defined by the following group presentation:


$$\langle x_0,x_1,x_2,\dots\mid x_j^{x_i}=x_{j+1}\ (j>i)\rangle,$$


where $a^b=b^{-1}ab$. It is generated by $x_0$, $x_1$. Let $C$ be the corresponding Cayley graph. We present the following results about $C$.

1) $C$ has a doubling property if and only if $\bar d(C)\le3$. (This is in fact true for any $2$-generated group.)

2) There exists a family of finite subgraphs $\Gamma_{n,m}$ in $C$ such that $\lim_{m\to\infty}d(\Gamma_{n,m})=6(n-1)/(2n-1)$. In particular, $\bar d(C)\ge\sup d(\Gamma_{n,m})=3$.

We conjecture that $d(C)=3$ for the Cayley graph of $F$ in generators $x_0$, $x_1$. If this conjecture is true, then $F$ is not amenable.

Besides, we present an easy formula to find the length of a given element $g\in F$ in generators $x_0$, $x_1$.





Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.

 

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