MAXIMAL AND POINTWISE ERGODIC-THEOREMS FOR WORD-HYPERBOLIC GROUPS

Let Gamma denote a word-hyperbole group, and let S = S-1 denote a fini te symmetric set of generators. Let S-n = {w : \w\ = n} denote the sph ere of radius n, where \ .\ denotes the word length on Gamma induced b y S. Define sigma(n) (d) double under bar (1/#S-n) Sigma(w epsilon Sn) w, and mu(n) = (1/(n + 1)) Sigma(k=0)(n)sigma(k) Let (X, B, m) be a p robability space on which Gamma acts ergodically by measure-preserving transformations. We prove a strong maximal inequality in L-2 for the maximal operator f(mu) = sup(n greater than or equal to 0)\mu(n)f(x)\ . The maximal inequality is applied to prove a pointwise ergodic theor em in L-2 for exponentially mixing actions of Gamma of the following f orm: mu(n)f(x) --> integral(x) f dm almost everywhere and in the L-2-n orm, for every f is an element of L-2(X). As a corollary, for a unifor m lattice Gamma subset of G, where G is a simple Lie group of real ran k one, we obtain a pointwise ergodic theorem for the action of Gamma o n an arbitrary ergodic G-space. In particular, this result holds when X = G/Lambda is a compact homogeneous space, and yields an equidistrib ution result for sets of lattice points of the form Gamma g, for almos t every g is an element of G.