  Bulk hybride halide perovskites are governed by significant Rashba and
Dresselhaus splitting. This indicates that such effects will not only affect
their optoelectronic properties but also those of their two dimensional layered
relatives. This work aims at understanding how different ways of symmetry
breaking influence these effects in those materials. For this purpose, model
structures are adopted where the organic compounds are replaced by Cs atoms.
Disregarding possible distortions in the inorganic layers, results in
structures with composition Cs$_{n+1}$Pb$_n$I$_{3n+1}$. Using the all-electron
full-potential density-functional-theory code \texttt{exciting}, the impact of
atomic displacement on the band structure is systematically studied for $n=1$,
2, 3 and $\infty$. The displacement patterns that yield Rashba or Dresselhaus
splitting are identified, and the amount of the splitting is determined as a
function of displacement. Furthermore, the spin textures in the electronic
states around the band gap are analyzed to differentiate between Rashba and
Dresselhaus effects. This study reveals in-plane Pb displacements as the origin
of the strongest effects.
