Construction of infinite finitely presented nilsemigroup

An infinite finitely presented nilsemigroup with identity $x^9$ = 0 is constructed. This construction answers the question of L.N. Shevrin and M.V. Sapir. The proof is based on the construction of a sequence of geometric complexes, each obtained by gluing several simple 4-cycles (squares). These complexes have certain geometric and combinatorial properties. Actually, the semigroup is the set of word codings of paths on such complexes. Each word codes a path on some complex. Defining relations correspond to pairs of equivalent short paths. The shortest paths in terms of the natural metric are associated with nonzero words in the subgroup. Codings that are not presented by some path or presented by non-shortest paths can be reduced to a zero word.