We establish a connection between extreme values of stable random fields arising in probability and groups G acting geometrically on CAT(-1) spaces X. The connection is mediated by the action of the group 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 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 is not a tree whose edges are (up to scale) integers. We also establish an analogous statement for normal subgroups of free groups. This is joint work with Parthanil Roy.