  We present a non-compact (4 + 1) dimensional model with a local strong
four-fermion interaction supplementing it with gravity. In the strong coupling
regime it reveals the spontaneous translational symmetry breaking which
eventually leads to the formation of domain walls, or thick 3-branes, embedded
in the AdS-5 manifold. To describe this phenomenon we construct the appropriate
low-energy effective Action and find kink-like vacuum solutions in the
quasi-flat Riemannian metric. We discuss the generation of ultra-low-energy (3
+ 1) dimensional physics and we establish the relation among the bulk five
dimensional gravitational constant, the brane Newton's constants and the
curvature of AdS-5 space-time. The plausible relation between the compositeness
scale of the scalar matter and the symmetry breaking scale is shown to support
the essential decoupling of branons, the scalar fluctuations of the brane, from
the Standard Model matter, supporting their possible role in the dark matter
saturation. The induced cosmological constant on the brane does vanish due to
exact cancellation of matter and gravity contributions.
