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.