Relative quasimorphisms and stably unbounded norms on the group of symplectomorphisms of the Euclidean spaces

In the paper where Burago–Ivanov–Polterovich defined the notion of conjugation-invariant norms on groups, they asked whether there exists a group with stably bounded commutator length admitting stably unbounded norms. We show that the kernel of the Calabi homomorphism of the group of symplectomorphisms of the even-dimensional Euclidean space with compact support is such a group. To prove its stable unboundedness, we consider quasimorphisms relative to a conjugation-invariant norm.

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Published 24 June 2016