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This paper is devoted to construction of finitely presented infinite nilsemigroup with identity $x^9=0$. This construction answers to the problem ofLev Shevrin and Mark Sapir. The paper is quite long so the proof is separatedinto geometric, combinatorial and finalization parts. In the first part weconstruct uniformly elliptic space. Space is called {\it uniformly elliptic} ifany two points $A$ and $B$ at the distance of $D$ can be connected by thesystem of geodesics which form a disc with width $\lambda\cdot D$ for someglobal constant $\lambda>0$. In the second part we study combinatorialproperties of the constructed complex. Vertices and edges of this complex codedby finite number of letters so we can consider semigroup of paths. Definingrelations correspond to pairs of equivalent short paths on the complex.Shortest path in sense of natural metric correspond nonzero words in thesemigroup. Words which are not presented as paths on complex and wordscorrespond to non shortest paths can be reduced to zero. In the third part wemake a finalization. In particular, we show that word containing ninth degreeword can be reduced to zero by defining relations. The present paper containsfirst part of the proof. This work was carried out with the help of the RussianScience Foundation Grant N 17-11-01377. The first author is the winner of thecontest Young Mathematics of Russia .

The complexes with property of uniform ellipticity