Growth of monomial algebras, simple rings and free subalgebras (Q1682048)

In the paper under review, the author considers functions \(f:{\mathbb N}\to{\mathbb N}\) which are monotonously increasing (\(f(n)<f(n+1)\) for all \(n\in\mathbb N\)) and submultiplicative (\(f(m+n)\leq f(m)f(n)\) for all \(m,n\in\mathbb N\)). It was shown in [\textit{A. Smoktunowicz} and \textit{L. Bartholdi}, Q. J. Math. 65, No. 2, 421--438 (2014; Zbl 1312.16017)] that if \(f\) grows sufficiently rapidly (there is \(\alpha>0\) such that \(nf(n)\leq f(\alpha n)\) for all \(n\in\mathbb N\)), then there is a finitely generated monomial algebra with growth type \(f\). The construction of Smoktunowicz and Bartholdi was modified by the author in [Isr. J. Math. 220, No. 1, 161--174 (2017; Zbl 1371.16019)] to produce a primitive algebra. In the present paper, this construction has been further modified to obtain algebras which are just-infinite, i.e., infinite dimensional algebras with the property that all proper quotients are finite dimensional. Combining this with ideas from [\textit{V. Nekrashevych}, Int. J. Algebra Comput. 26, No. 2, 375--397 (2016; Zbl 1366.16016)], the author constructs finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. The constructions are determined in terms of infinite words which gives rise to infinite uniformly recurrent words with subword complexity having the same growth type. It is known that uniformly recurrent words are important objects from combinatorial, dynamical, and algebraic points of view. Then the author studies the entropy of algebras. If \(R=\bigoplus_{i\in\mathbb N}R_i\) is a graded algebra with \(\dim R_i<\infty\), then the entropy is defined by \[ H(R)=\limsup_{n\to\infty}\root n\of{\dim\bigoplus_{i\leq n}R_i}. \] The algebras in the paper have sharp entropy. (The proper quotients have strictly smaller entropy.) If \(R\) contains a free subalgebra, then \(H(R)>1\). The author obtains an evident better lower bound on \(R\) in terms of the degree of the homogeneous components on which the free generators are supported. He shows that this evident bound cannot be significantly improved: there exists a family of graded algebras with entropy arbitrarily close to 1 with free subalgebras with generators in the expected degrees.