  We consider the compactification of a Yang-Mills theory on a
three-dimensional nilmanifold. The compactification generates a Yang-Mills
theory in four space-time dimensions, coupled to a specific scalar sector. The
compactification geometry gives rise to masses for the zero-modes, proportional
to the twist parameter of the nilmanifold. We study the simple example of an SU
(3) model broken by a non-trivial vacuum of the scalar potential which
generates three mass scales, two being at tree level, and the third one at loop
level. We point out the relevance of general twisted geometries for model
building and in particular for gauge-Higgs type models, as the twist generates
tree-level mass hierarchies useful for grand unification and for the Higgs
sector in electroweak symmetry breaking.
