In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces and of CAT (0)-groups. We show that, for mapping class groups of surfaces, these functions exhibit phase transitions at the rank (as measured by 3·genus+number of punctures−3). We also provide inductive constructions of CAT (0)-spaces with co-compact group actions, for which the divergence below the rank is (exactly) a polynomial function of our choice, with degree arbitrarily large compared to the dimension.

The conference was in the area of geometric group theory, the field of mathematics in which one studies infinite groups (finitely generated, or more generally locally compact, countable etc.) via actions on spaces endowed with various structures (geometric, measurable, analytic etc.). The surging current activity in the field is drawing more and more connections with other mathematical areas, and this was successfully reflected in the program of this week, during which problems in algebraic topology, representation theory and functional analysis, to name just a few, featured prominently alongside core topics in the area.

The conference focused on the use of geometric methods to study infinite groups and the interplay of group theory with other areas. One of the central techniques in geometric group theory is to study infinite discrete groups by their actions on nice, suitable spaces. These spaces often carry an interesting large-scale geometry, such as non-positive curvature or hyperbolicity in the sense of Gromov, or are equipped with rich geometric or combinatorial structure. From these actions one can investigate structural properties of the groups. This connection has become very prominent during the last years. In this context non-discrete topological groups, such as profinite groups or locally compact groups appear quite naturally. Likewise, analytic methods and operator theory play an increasing role in the area.