CHAPTER 1

THE GREEN FUNCTION

1. Random Walks on a Tree

A graph is defined as a pair (V, J5), where V is a set of vertices and E1 a family

of unordered pairs of vertices, called edges . A graph is said to be locally finite if

each element x of V belongs to finitely many edges. A tree is a graph with two

special properties: (1) it is connected and (2) it has no circuits. To define these

properties we must introduce the notion of chain: a chain is a finite sequence of

distinct vertices {xo,X\, • • • ,x

m

} such that

{XJ-I,XJ}

€ E, for 1 j m. We

say that the graph is connected if given any two distinct vertices x and y there

exists a chain {xo,#i, • * •

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#m} such that x = xo and y = xm. In this case we

also say that the chain joins x and y. The graph has no circuits if given two

vertices x and y there exists at most one chain joining x and y.

In this paper we are only concerned with locally finite homogeneous trees. A

tree is called homogeneous of degree d if each vertex belongs to exactly d edges.

For a locally finite tree, the degree must be a positive integer which we will

always denote by q-\-1. If q = 0, then the tree has two vertices; if q = 1, then the

tree is one doubly infinite chain of vertices. Figures la and lb show homogeneous

trees of degrees three and four.

The vertex set of any tree has a natural metric space structure. Let x, y G V

and let {x =

XQ,X\,

• • • ,x

m

= y} be the unique chain joining x and y. Then

m is the distance between x and y denoted d(x,y). (Of course this definition

makes d(x,x) = 0.)

We shall be concerned with a certain group G whose Cayley graph is a tree.

If G is any group and #i, • • • , xn a given set of generators, the Cayley graph of

G, with respect to these generators, has vertex set V equal to G and edge set

E = {{X,XXJ}; x £ G,j = 1, • • -n} .

Let G be the free product G = Z2 * • • • * Z2 of q + 1 copies of Z2, and let

ai, • • • , a

q +

i be the obvious generators of G, each one an involution generating

one copy of Z2. The Cayley graph of G with respect to these generators is a

homogeneous tree of degree q + 1. For q+ 1 = 3 this tree is represented by the

diagram of Figure la (where we denote ai, a2, as by a, 6, and c.)

Other groups also have Cayley graphs which are trees. For instance if G is a

free group on two generators a and b, then the Cayley graph of G with respect

to the generators a and b is a tree represented by the diagram of Figure lb. The

results and methods of this paper apply, in fact, to all groups having a Cayley

graph which is a homogeneous tree. However, for clarity we will consider only

the group G = Z2 * • • • • Z2.

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