Growth, unavoidable words, and M. Sapir's conjecture for semigroup varieties.
Let \(u_1,u_2,\dots,u_n,\dots\) be a countable set of variables. Put \(Z_1\equiv u_1\) and \(Z_{n+1}\equiv Z_nu_{n+1}Z_n\) for any integer \(n>1\). In 1993, M.~Sapir posed the conjecture that if a nonperiodic semigroup variety \(\mathcal R\) satisfies a nontrivial identity of the type \(Z_n=W\) for some \(n>1\) then every finitely generated semigroup in \(\mathcal R\) has subexponential growth. The main result of the article under review (Theorem 1.1) states that the conjecture is true. An infinite word \(W\) is called uniformly recurrent if for every finite subword \(U\) of \(W\) there exists a number \(N\) such that every subword of \(W\) of length \(N\) contains \(U\) as a subword. Let \(S\) be a semigroup with a finite set \(X\) of generators and \(X^+\) the free semigroup over \(X\). Let \(\preceq\) denote the lexicographic order in \(X^+\). The word \(U\in X^+\) is called shortlex reduced relative to a semigroup \(S\) if for any word \(V\in X^+\) such that \(U=V\) in \(S\), we have \(\ell(U)\leq \ell(V)\) and if \(\ell(U)=\ell(V)\) then \(U\preceq V\) in \(X^+\) (where \(\ell(W)\) denotes the length of the word \(W\)). An infinite word \(\omega\) is called shortlex reduced relative to \(S\) if any finite subword of \(\omega\) is. The second result of the article (Theorem 1.2) states that if a nonperiodic semigroup variety \(\mathcal R\) satisfies a nontrivial identity of the type \(Z_n= W\) for some \(n>1\) and \(S\) is a finitely generated semigroup from \(\mathcal R\) then every infinite word which is uniformly recurrent and shortlex reduced relative to \(S\) is periodic and the length of this period is bounded above by some constant depending only on \(n\) and on the number of generators of \(S\). As applications of these results, the author constructs examples of semigroup varieties for which every noncyclic relatively free semigroup has intermediate growth and gives examples of relatively free semigroups of intermediate growth with superdimension 1.
subexponential growth, intermediate growth
scientific article; zbMATH DE number 2037100
scientific article