The complexes with property of uniform ellipticity

TL;DR

This paper constructs a finitely presented infinite nil semigroup with a specific nilpotency property, using geometric and combinatorial methods to demonstrate uniform ellipticity and reduction rules.

Contribution

It introduces the concept of uniformly elliptic complexes and applies this to construct a nil semigroup with a ninth-degree nilpotency, answering a problem posed by Shevrin and Sapir.

Findings

Construction of a uniformly elliptic space.

Development of a semigroup of paths with defining relations.

Proof that ninth-degree words reduce to zero.

Abstract

This paper is devoted to construction of finitely presented infinite nil semigroup with identity $x^9=0$. This construction answers to the problem of Lev Shevrin and Mark Sapir. The paper is quite long so the proof is separated into geometric, combinatorial and finalization parts. In the first part we construct uniformly elliptic space. Space is called {\it uniformly elliptic} if any two points $A$ and $B$ at the distance of $D$ can be connected by the system of geodesics which form a disc with width $\lambda\cdot D$ for some global constant $\lambda>0$. In the second part we study combinatorial properties of the constructed complex. Vertices and edges of this complex coded by finite number of letters so we can consider semigroup of paths. Defining relations correspond to pairs of equivalent short paths on the complex. Shortest path in sense of natural metric correspond nonzero words in the semigroup. Words which are not presented as paths on complex and words correspond to non shortest paths can be reduced to zero. In the third part we make a finalization. In particular, we show that word containing ninth degree word can be reduced to zero by defining relations. The present paper contains first part of the proof. This work was carried out with the help of the Russian Science Foundation Grant N 17-11-01377. The first author is the winner of the contest Young Mathematics of Russia .