Higher dimensional divergence for mapping class groups

TL;DR

This paper studies the higher dimensional divergence functions of mapping class groups and CAT(0)-groups, revealing phase transitions and constructing spaces with customizable polynomial divergence behaviors.

Contribution

It demonstrates phase transitions in divergence functions of mapping class groups and provides methods to construct CAT(0)-spaces with prescribed polynomial divergence below a certain rank.

Findings

Divergence functions show phase transitions at the rank for mapping class groups.

Constructed CAT(0)-spaces with polynomial divergence of arbitrary degree below the rank.

Divergence behaviors can be precisely controlled in constructed spaces.

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 thrice the genus plus the number of punctures minus 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.