disconnected (Fig. \[fig:bubblestrip\], right). CDT quantum gravity in 1+1 dimensions has only strips, whereas the version without distinguished foliation has both strips and bubbles. Analogous structures will play a role in our analysis in 2+1 dimensions too, where also the interior structure of a bubble can become more complicated.

Relaxing the foliation: Kinematics in 2+1 dimensions
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### Local causality conditions {#sec:nfct21 .unnumbered}

Following an analogous procedure in 2+1 dimensions to arrive at a model without distinguished simplicial hypermanifolds, we first must determine which flat tetrahedra – again only built from two types of edge lengths – give rise to Minkowskian building blocks of the correct Lorentzian signature $(-++)$. Fig. \[fig:tetrahedra\] shows all types of tetrahedra which can be constructed using space- and timelike links with fixed squared lengths $\ell_s^2$ and ${\ell_t^2\! =\! -\alpha \ell_s^2}$ respectively. In the rest of this document we will set $\ell_s\! =\! 1$. By calculating the metric in the interior of each tetrahedron type, one finds that only $T_2$, $T_3$, $T_5$ and $T_9$ have the required signature for all values $\alpha\! >\! 0$ of the asymmetry parameter, and type $T_7$ only for $0\! <\! \alpha\! <\! 1$. Note that standard CDT quantum gravity only uses 