We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on a suitable space $X$. The connection between extreme values and the indexing group $G$ is mediated by the action of $G$ on the 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 quantifies the distortion of measures on the boundary in comparison to the movement of points in the space $X$. We 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. (Joint work with Jayadev Athreya and Mahan Mj, under review in Probability Theory and Related Fields.)

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Last updated: 03 Feb 2023