Split quasicocycles

Abstract

Let E be a linear isometric representation of a group \Gamma. In this paper we construct and study a family of quasicocycles \Gamma -> E that arise from splittings \Gamma = A * B. Under certain assumptions on A, B and E the bounded cohomology classes associated to these quasicocycles form an infinite-dimensional subspace of H^2_b(\Gamma,E). This is in particular the case when \Gamma is free and E finite-dimensional or of the type l^p(\Gamma). For the trivial target E = R we obtain a new family of quasimorphisms for which we compute the Gromov norm in bounded cohomology. This yields a linear isometric embedding D(A) \oplus D(B) -> H^2_b(\Gamma,R), where D(A) is a Banach space which is norm-equivalent to the alternating subspace of l^inf(A). We prove that there are classes of our type in H^2_b(F_2,R) which have infinite stabilizer under the natural action of Out(F_2). By replacing the target E with a group G with bi-invariant metric we obtain a new type of quasi-representations \Gamma -> G that arise from splittings of \Gamma.