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Characterising actions on trees yielding non-trivial quasimorphisms. (English) Zbl 07341610

Summary: Using a cocycle defined by Monod and Shalom [J. Differ. Geom. 67, No. 3, 395–455 (2004; Zbl 1127.53035)] 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 \(\infty\)-transitive.

MSC:

20E08 Groups acting on trees
55N35 Other homology theories in algebraic topology
20F65 Geometric group theory

Citations:

Zbl 1127.53035
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References:

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