  At the heart of the study of topological insulators lies a fundamental
dichotomy: topological invariants are defined in infinite systems, but surface
states as their main footprint only exist in finite systems. In the slab
geometry, namely infinite in two planar directions and finite in the
perpendicular direction, the 2D topological invariant was shown to display
three different types of behaviour. The perpendicular Dirac velocity turns out
to be a critical control parameter discerning between different qualitative
situations. When it is zero, the three types of behaviour extrapolate to the
three 3D topologically distinct phases: trivial, weak and strong topological
insulators. We show analytically that the boundaries between types of behaviour
are topological phase transitions of particular significance since they allow
to predict the 3D topological invariants from finite-thickness transitions.
When the perpendicular Dirac velocity is not zero, we identify a new phase with
surface states but no band inversion at any finite thickness, disentangling
these two concepts which are closely linked in 3D. We also show that at zero
perpendicular Dirac velocity the system is gapless in the 3D bulk and therefore
not a topological insulating state, even though the slab geometry extrapolates
to the 3D topological phases. Finally, in a parameter regime with strong
dispersion perpendicular to the surface of the slab, we encounter the unusual
case that the slab physics displays non-trivial phases with surface states but
nevertheless extrapolates to a 3D trivial state.
