  Energy deposition by active galactic nuclei jets into the ambient medium can
affect galaxy formation and evolution, the cooling of gas flows at the centres
of galaxy clusters, and the growth of the supermassive black holes. However,
the processes that couple jet power to the ambient medium and determine jet
morphology are poorly understood. For instance, there is no agreement on the
cause of the well-known Fanaroff-Riley (FR) morphological dichotomy of jets,
with FRI jets being shorter and less stable than FRII jets. We carry out global
3D magnetohydrodynamic simulations of relativistic jets propagating through the
ambient medium. We show that the flat density profiles of galactic cores slow
down and collimate the jets, making them susceptible to the 3D magnetic kink
instability. We obtain a critical power, which depends on the galaxy core mass
and radius, below which jets become kink-unstable within the core, stall, and
inflate cavities filled with relativistically-hot plasma. Jets above the
critical power stably escape the core and form powerful backflows. Thus, the
kink instability controls the jet morphology and can lead to the FR dichotomy.
The model-predicted dependence of the critical power on the galaxy optical
luminosity agrees well with observations.
