Characterising actions on trees yielding non-trivial quasimorphisms

TL;DR

This paper characterizes actions on trees that produce non-trivial median quasimorphisms, linking group actions, boundary behavior, and cohomology, with applications to lattices in products of trees.

Contribution

It fully characterizes when actions on trees yield non-trivial median quasimorphisms, revealing conditions for infinite-dimensional second bounded cohomology.

Findings

Actions with high geodesic transitivity or boundary fixation produce trivial quasimorphisms.

Existence of infinite families of median quasimorphisms correlates with infinite-dimensional cohomology.

Cocompact lattices in products of trees have trivial quasimorphisms under local transitivity conditions.

Abstract

We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded cohomology of the group is infinite dimensional as a vector space. As an application, we show that a cocompact lattice in a product of trees only has trivial quasimorphisms if and only if both closures of the projections on the two factors are locally $\infty$-transitive.