work which considered explicitly a possible relaxation of CDT’s strict time slicing.[^4] A soft way of relaxing the foliation in 1+1 dimensions was studied in [@markopoulousmolin], where under certain conditions the timelike links were allowed to have varying length. In this approach the foliation is still present, since the connectivity of the underlying triangulation is unchanged, but the individual leaves of the foliation are not placed at equidistant intervals. The authors argued that this should not affect the continuum limit of the model. A similar idea in 2+1 dimensions was considered in [@konopka], where it was also suggested to add new elementary building blocks to CDT, which extend over two slabs of the foliation instead of one. This study did not include details of how the path integral of the corresponding generalization of CDT quantum gravity should be formulated or simulated.

The remainder of this article is structured as follows. We begin with a brief review of Causal Dynamical Triangulations in Sec. \[cdt\]. In Sec. \[sec:ncdt\_2d\] we discuss aspects of CDT without preferred foliation in 1+1 dimensions, to illustrate the basic geometric idea behind the enlarged model. Sec. \[sec:nfct21\] contains a detailed study of the kinematical aspects of nonfoliated CDT 