Growth, unavoidable words, and M. Sapirʹs conjecture for semigroup varieties

Let be a nonperiodic semigroup variety satisfying the nontrivial identity Zn=W, where Zn is the nth element of the so-called Ziminʹs sequence Z1=u1,Zn+1=Znun+1Zn (n=1,2,…) of unavoidable words. It is shown that every finitely generated (f.g.) semigroup has subexponential growth and every uniformly recurrent infinite word which is geodesic and lexicographically reduced relative to S is periodic. Furthermore, the length of the period is bounded above by a constant which depends only on the number n and on the number of generators of S. The first of these results gives a positive answer to the problem posed by M. Sapir. Some applications including examples of f.g. relatively free semigroups having intermediate growth and maximal (equals 1) superdimension are considered.