On Sandon-type metrics for contactomorphism groups

TL;DR

This paper introduces a new conjugation-invariant norm on the universal cover of contactomorphism groups for certain contact manifolds, revealing geometric properties and conditions for boundedness.

Contribution

It constructs a novel Sandon-type metric for contactomorphism groups and explores its properties and implications for contact geometry.

Findings

The norm admits a quasi-isometric embedding of the real line.

The norm descends to the contactomorphism group, preserving invariance.

Conditions for boundedness of conjugation-invariant norms are discussed.

Abstract

For certain contact manifolds admitting a 1-periodic Reeb flow we construct a conjugation-invariant norm on the universal cover of the contactomorphism group. With respect to this norm the group admits a quasi-isometric monomorphism of the real line. The construction involves the partial order on contactomorphisms and symplectic intersections. This norm descends to a conjugation-invariant norm on the contactomorphism group. As a counterpoint, we discuss conditions under which conjugation-invariant norms for contactomorphisms are necessarily bounded.