The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms

TL;DR

This paper explores the large-scale geometric properties of the group of area-preserving diffeomorphisms on surfaces using compactification techniques, establishing new embeddings and Lipschitz properties for related groups and quasi-morphisms.

Contribution

It introduces a novel application of the Schwarz-Milnor lemma to configuration spaces, enabling new quasi-isometric embeddings and Lipschitz bounds for groups related to surface diffeomorphisms.

Findings

All right-angled Artin groups embed quasi-isometrically into the diffeomorphism group.

Gambaudo-Ghys quasi-morphisms from braid groups are Lipschitz on this metric group.

The approach applies the Fulton-MacPherson compactification to configuration spaces for the first time.

Abstract

We prove a number of new results on the large-scale geometry of the $L^p$-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of $n$ points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving diffeomorphisms endowed with the $L^p$-metric, and that all Gambaudo-Ghys quasi-morphisms on this metric group coming from the braid group on $n$ strands are Lipschitz. This was conjectured to hold, yet proven only for low values of $n$ and the genus $g$ of the surface.