Effective quasimorphisms on free chains

TL;DR

This paper constructs effective homogeneous counting quasimorphisms on free groups, providing new insights into stable commutator length and the structure of quasimorphisms.

Contribution

It introduces specific quasimorphisms that detect chains in free groups and establishes a basis for the space of such quasimorphisms.

Findings

Homogeneous counting quasimorphisms effectively detect chains in free groups.

If a group has a free subgroup of index d, elements have stable commutator length ≥ 1/8d or relate to their inverse.

The space of homogeneous quasimorphisms on a finitely-generated free group has a countable basis.

Abstract

We find homogeneous counting quasimorphisms that are effective at seeing chains in a free group F. As corollary, we derive that if a group G has an index-d free subgroup, then every element g in G either has stable commutator length at least 1/8d or some power of g is conjugate to its inverse. We also show that for a finitely-generated free group F, there is a countable basis for the real vector space of homogeneous quasimorphisms on F.