Higher dimensional divergence for mapping class groups

Abstract

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.

Combinatorial higher dimensional isoperimetry and divergence

“…Theorem 5.3 is a powerful tool which we expect will be widely used. Indeed, in [BD2] the authors use this to construct CAT(0) groups with higher dimensional divergence exactly a polynomial function of our choice.…”