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. This solves a problem of Lev Shevrin and Mark Sapir. In this first part we obtain a sequence of complexes formed of squares (-cycles) having the following geometric properties. 1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant λ > 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 λ 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 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.