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 PattersonSullivan 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 nonvanishing is equivalent to finiteness of the BowenMargulis measure for the associated unit tangent bundle U(X/G) provided X/G has nonarithmetic length spectrum. As a consequence, we establish a dichotomy for the growthrate of a partial maxima sequence of stationary symmetric αstable (0<α<2) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups. This joint work with Mahan MJ and Parthanil Roy.
Stable Random Fields, PattersonSullivan measures and Extremal Cocycle Growth
Jayadev Athreya, UW

THO 125
We study extreme values of groupindexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: