Cogrowth of Regular Graphs

Let & be a ¿-regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S . As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on 9 is zero. Grigorchuk's criterion for amenability of finitely generated groups follows. In this note, we shall relate the first eigenvalue of the Laplacian on a connected regular graph to the size of the kernel of the universal covering map. The main results have been proven in [C, G, P]. The proof presented here appears simpler; it depends on the explicit formula for minimal positive solutions of AF + eF = -I. Let f be a connected simple graph with constant vertex degree d > 3, F be the universal covering tree of "§, and 6 the covering map (i.e., 6 is a vertex surjection of T on ^ that preserves adjacency and vertex degree). We let T and & denote the vertex sets of the corresponding graphs. Note that F has constant vertex degree d. Since F is connected, F may be considered a metric space with the usual graph metric a (S(x, y) is the length of the shortest path connecting x and y). For x £ T and zz > 0, let [x] = 6~x(6(x)) and Sn(x) = {y: S(x, y) = n} . For x, y £ T, note that limsup|[y]n5„(x)|1/n = infiA>0: V *-*(*• *) < oo I «6W and is thus independent of x and y. We call this number, cogr(F, &), the cogrowth constant of & in T. For x, y vertices of a graph, we write xEy if x and y are connected by an edge. For x, y £ T, let _ / i if x£y ' \ 0 otherwise. Note that q is the transition matrix of the simple random walk on F. Let a»") denote the nth power of q . For a, b £ 3? and x £ 8~x(a), since 6 takes the Received by the editors January 25, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 60J15; Secondary 05C05, 05C25, 43A07.