  The original models of causal dynamical triangulations construct space-time
by arranging a set of simplices in layers separated by a fixed time-like
distance. The importance of the foliation structure in the 2+1 dimensional
model is studied by considering variations in which this property is relaxed.
It turns out that the fixed-lapse condition can be equivalently replaced by a
set of global constraints that have geometrical interpretation. On the other
hand, the introduction of new types of simplices that puncture the foliating
sheets in general leads to different low-energy behavior compared to the
original model.
