We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.
Brandenbursky, M., Gal, S., Kedra, J., & Marcinkowski, M. (2016). Cancellation norm and the geometry of bi-invariant word metrics. Glasgow Mathematical Journal, 58(01), 153-176. https://doi.org/10.1017/S0017089515000129