Finitely presented nilsemigroups: complexes with the property of uniform ellipticity

• Published in: Izvestiya. Mathematics 2021-12, Vol.85 (6), p.1146-1180
• Main Authors: Ivanov-Pogodaev, I. A., Kanel-Belov, A. Ya
• Format: Article
• Language: English
• Subjects: Apexes; Ellipticity; finitely presented groups; finitely presented rings; finitely presented semigroups; Graph theory; nilsemigroups
• ISSN: 1064-5632; 1468-4810
• DOI: 10.1070/IM8978

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 such that in the set of shortest paths of length connecting points and there are two paths such that the distance between them is at most . 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 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.