Construction of infinite finitely presented nilsemigroup

TL;DR

This paper constructs an infinite finitely presented nilsemigroup by leveraging aperiodic tilings and properties of uniformly elliptic spaces, addressing a longstanding problem in algebraic semigroup theory.

Contribution

It provides the first explicit construction of an infinite finitely presented nilsemigroup, solving the Shevrin and Sapir problem using geometric and tiling methods.

Findings

Constructed an infinite finitely presented nilsemigroup.

Applied aperiodic tilings and elliptic space properties.

Resolved a major open problem in algebraic semigroup theory.

Abstract

The paper deals with the solution of Shevrin ans Sapir problem. Infinite finitely presented nilsemigroup is constructed. The construction is based on aperiodic tilings, Goodman-Strauss type theorems on uniformly elliptic space. Space is called {\it uniformly elliptic} iff there is universal constant $\lambda>0$ such that any two points $A$ and $B$ on distant $D$ can be joined by family of geodesic lines generating a disc of wideness $\lambda\cdot D$. Research was supported by the grant RFBR N 14-01-00548