Growth of linear semigroups. II (Q1906637)

Let \(S\subseteq M_n(K)\) be a finitely generated linear semigroup. The author gave [in part I, J. Aust. Math. Soc., Ser. A 59, 1-13 (1995), Linear semigroups of polynomial growth in positive characteristic, J. Pure Appl. Algebra (to appear)]\ a complete description of such semigroups \(S\) of polynomial growth in the case \(\text{ch}(K)>0\) and some sufficient condition in the general case. Now he completes the description in the case where \(\text{ch}(K)= 0\). The main result states that \(S\) has polynomial growth if and only if (1) the groups associated to \(S\) are finitely generated and almost nilpotent and (2) for every uniform component \(U\) of \(S\), a certain semigroup derived from \(S\) by means of \(U\) is almost unipotent. Here the groups associated with \(S\) are groups generated by intersections \(S\cap D\) with maximal subgroups \(D\) of \(M_n(K)\), and an almost unipotent semigroup is a semigroup whose nonempty intersections with maximal subgroups of \(M_n(K)\) generate almost unipotent subgroups.