Finitely presented nilsemigroups: complexes with the property of uniform ellipticity

This paper is the first in a series of three devoted to constructing a finitely presented infinite nilsemigroup satisfying the identity $x^9=0$. This solves a problem of Lev Shevrin and Mark Sapir.
In this first part we obtain a sequence of complexes formed of squares ($4$-cycles) having the following geometric properties.
1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant $\lambda>0$ such that in the set of shortest paths of length $D$ connecting points $A$ and $B$ there are two paths such that the distance between them is at most $\lambda D$. In this case, the distance between paths with the same beginning and end is defined as the maximal distance between the corresponding points.
2) Complexes are nested. A complex of level $n+1$ is obtained from a complex of level $n$ by adding several vertices and edges according to certain rules.
3) Paths admit local transformations. Assume that we can transform paths by replacing a path along two sides of a minimal square by the path along the other two sides. Two shortest paths with the same ends can be transformed into each other locally if these ends are vertices of a square in the embedded complex.
The geometric properties of the sequence of complexes will be further used to define finitely presented semigroups.