FIXED-POINTS OF AUTOMORPHISMS OF FREE PRO-P GROUPS OF RANK-2

Let p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism alpha of F of finite order m, and let Fix(F)(alpha) = {x is an element of F \ alpha(x) = x} be the subgroup of F consisting of the elements fixed by alpha. It is known that if m is prime to p and alpha = id(F), then the rank of Fix(F)(alpha) is infinite. In this paper we show that if m is a finite power p(r) of p, the rank of Fix(F)(alpha) is at most 2. We conjecture that if the rank off is n and the order of alpha is a power of p, then rank(Fix(F)(alpha)) less than or equal to n.