Group actions on twisted sums of Banach spaces

We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$-centralizers; and that, under (A) and (U), $G$-centralizers are bounded perturbations of $G$-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of $L_p(0,1), p\neq 2$ on the Kalton-Peck space $Z_p$, certain non-unitarizable triangular representations of the free group $F_\infty$ on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of $L_p$-spaces. In the last section we consider the category of $G$-Banach spaces and study its exact sequences, showing that, under (A) and (U), $G$-splitting and usual splitting coincide.