Linear semigroups of polynomial growth in positive characteristic
The problem of polynomial growth for linear semigroups (over fields) attracted attention of different mathematicians. In the present paper this problem is solved completely for any finitely generated linear semigroup over a field of positive characteristic. Let \(K\) be a field, \(\text{ch }K>0\), and let \(M_n(k)\) be the full matrix semigroup over \(K\). The main result of the paper is the following. Suppose \(S\) is a finitely generated subsemigroup of \(M_n(K)\). Then \(S\) has polynomial growth if and only if every nonempty intersection \(S\cap D\) with a maximal subgroup \(D\) of \(M_n(K)\) has a finitely generated nilpotent-by-finite group of fractions. Moreover, the degree of growth of such \(S\) is bounded by a function \(f(n,r)\) of \(n\) and of the maximal rank \(r\) of a free commutative subsemigroup of \(S\). For \(K\) an arbitrary field, the author published analogous (but not so complete) results [in J. Aust. Math. Soc., Ser. A 60, No. 1, 18-30 (1996; Zbl 0852.20055)].
finitely generated nilpotent-by-finite group of fractions
scientific article; zbMATH DE number 908492
scientific article