TY - JOUR

T1 - Characterising actions on trees yielding non-trivial quasimorphisms

AU - Iozzi, Alessandra

AU - Pagliantini, Cristina

AU - Sisto, Alessandro

PY - 2020/5/19

Y1 - 2020/5/19

N2 - Using a cocycle defined by Monod and Shalom (J Differential Geom 67(3):395–455, 2004) we introduce the median quasimorphisms for groups acting on trees. Then we characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, or 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 the automorphism group of a product of trees has only trivial quasimorphisms if and only if the closures of the projections on each of the two factors are locally ∞-transitive.

AB - Using a cocycle defined by Monod and Shalom (J Differential Geom 67(3):395–455, 2004) we introduce the median quasimorphisms for groups acting on trees. Then we characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, or 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 the automorphism group of a product of trees has only trivial quasimorphisms if and only if the closures of the projections on each of the two factors are locally ∞-transitive.

KW - Actions on trees

KW - Bounded cohomology

KW - Quasimorphisms

UR - http://www.scopus.com/inward/record.url?scp=85085282024&partnerID=8YFLogxK

U2 - 10.1007/s40316-020-00137-3

DO - 10.1007/s40316-020-00137-3

M3 - Article

AN - SCOPUS:85085282024

SN - 2195-4755

JO - Annales mathématiques du Québec

JF - Annales mathématiques du Québec

ER -