On the Properties of The Cayley Graph of Richard Thompson's Group F
نویسنده
چکیده
We study some properties of the Cayley graph of the 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 2generated 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 into the 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 graph does not exceed 3 if and only if the group satisfies the same 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 to find the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper we show that the lower bound for the growth rate of F is (3 + √ 5)/2.
منابع مشابه
Cayley graph associated to a semihypergroup
The purpose of this paper is the study of Cayley graph associated to a semihypergroup(or hypergroup). In this regards first we associate a Cayley graph to every semihypergroup and then we study theproperties of this graph, such as Hamiltonian cycles in this graph. Also, by some of examples we will illustrate the properties and behavior of these Cayley graphs, in particulars we show that ...
متن کاملCOMPUTING THE EIGENVALUES OF CAYLEY GRAPHS OF ORDER p2q
A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.
متن کاملSOME GRAPH PARAMETERS ON THE COMPOSITE ORDER CAYLEY GRAPH
In this paper, the composite order Cayley graph Cay(G, S) is introduced, where G is a group and S is the set of all composite order elements of G. Some graph parameters such as diameter, girth, clique number, independence number, vertex chromatic number and domination number are calculated for the composite order Cayley graph of some certain groups. Moreover, the planarity of composite order Ca...
متن کاملOn trivial ends of Cayley graph of groups
In this paper, first we introduce the end of locally finite graphs as an equivalence class of infinite paths in the graph. Then we mention the ends of finitely generated groups using the Cayley graph. It was proved that the number of ends of groups are not depended on the Cayley graph and that the number of ends in the groups is equal to zero, one, two, or infinity. For ...
متن کاملOn the eigenvalues of normal edge-transitive Cayley graphs
A graph $Gamma$ is said to be vertex-transitive or edge- transitive if the automorphism group of $Gamma$ acts transitively on $V(Gamma)$ or $E(Gamma)$, respectively. Let $Gamma=Cay(G,S)$ be a Cayley graph on $G$ relative to $S$. Then, $Gamma$ is said to be normal edge-transitive, if $N_{Aut(Gamma)}(G)$ acts transitively on edges. In this paper, the eigenvalues of normal edge-tra...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- IJAC
دوره 14 شماره
صفحات -
تاریخ انتشار 2004