Pascal Rolli: Split quasimorphisms

We observe that on a free product of groups there is a natural family of quasimorphisms obtained by extension of quasimorphisms on the free factors. This simple construction yields non-trivial quasimorphisms even if the maps on the factors are trivial. As a consequence we obtain an embedding of \ell^\infty into the second bounded cohomology of a non-abelian free group. We discuss the homogenization and the Gromov norm of these split quasimorphisms and introduce some generalizations, namely split quasicocycles into Hilbert spaces and split epsilon-representations.