  In this work, we solve an open problem mentioned in Xiao (2010) and provide
sharp bounds on the rate of growth of maximal moments for stationary symmetric
stable random fields using structure theorem of finitely generated abelian
groups and ergodic theory of quasi-invariant group actions. We also investigate
the relationship between this rate of growth and the path regularity properties
of self-similar stable random fields with stationary increments, and establish
uniform modulus of continuity of such fields. In the process, a new notion of
weak effective dimension is introduced for stable random fields and is
connected to maximal moments and path properties. Our results establish a
boundary between shorter and longer memory in relation to Holder continuity of
S alpha S random fields confirming a conjecture of Samorodnitsky (2004).
