  We study extreme values of group-indexed stable random fields for discrete
groups $G$ acting geometrically on spaces $X$ in the following cases:
  1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space
$X$,
  2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space
$X$,
  3) $G$ is the mapping class group of a surface acting on its Teichmuller
space.
  The connection between extreme values and the geometric action is mediated by
the action of the group $G$ on its limit set equipped with the
Patterson-Sullivan measure. Based on motivation from extreme value theory, we
introduce an invariant of the action called extremal cocycle growth which
measures the distortion of measures on the boundary in comparison to the
movement of points in the space $X$ and show that its non-vanishing is
equivalent to finiteness of the Bowen-Margulis measure for the associated unit
tangent bundle $U(X/G)$ provided $X/G$ has non-arithmetic length spectrum. As a
consequence, we establish a dichotomy for the growth-rate of a partial maxima
sequence of stationary symmetric $\alpha$-stable ($0 < \alpha < 2$) random
fields indexed by groups acting on such spaces. We also establish analogous
results for normal subgroups of free groups.
