Growth of varieties of groups and group representations, and the Gel'fand-Kirillov dimension (Q1903028)

This paper considers the growth rate of various varieties of groups and group representations in terms of the growth of the free algebras. Let \(\mathcal H\) be a variety of group representations. If \(\mathcal H\) is locally finite-dimensional, then the growth rate can be described in terms of the function \(f_{\mathcal H} (n)=\dim_K (KF_n/I_n)\), where \(KF_n/I_n\) is the free representation of rank \(n\) in \(\mathcal H\). If \(\mathcal H\) is not locally finite-dimensional, we have instead to use the Gel'fand-Kirillov dimension, and consider the function \(g_{\mathcal H} (n)=\text{GK} \dim(KF_n/I_n)\), where for any finitely-generated unital module \(A\) over a field \(K\), \(\text{GK}\dim(A)=\overline{\lim}_m (\log_m d_V(m))\), where \(V\) is any finite-dimensional generating subspace of \(A\) and \(d_m(V)=\dim_K(K \cdot 1 + V^2 + \dots + V^m)\). We may then define the growth, \(\text{gr}({\mathcal H})\) of \(\mathcal H\) to be \(\overline{\lim}_n\log_n (g_{\mathcal H})\). (Note that this function is not defined for locally finite-dimensional varieties, where it is more natural to use the function \(f_{\mathcal H} (n)\) -- these varieties are not considered in the rest of the paper.) Two problems which now arise are those of characterising the \(\mathcal H\) such that \(g_{\mathcal H}(n)\) is finite for all \(n\), and determining which real numbers can arise as values of \(\text{gr}({\mathcal H})\). These problems are solved completely for two particular sorts of varieties; those of group type and those of ring type. Let \(\mathcal V\) be a variety of groups and denote by \(\omega\mathcal V\) the class of all representations \(\rho=(V,G)\) such that \(G/\ker \rho \in \mathcal V\). Then \(\omega\mathcal V\) is a variety of group type. The definition of the varieties of ring type is rather too technical to give here. The results show that for varieties of group type the function \(g\) grows either polynomially or exponentially, whereas for those of ring type \(g_{\mathcal H} (n)\) is finite for every \(n\), and if \(K\) has characteristic 0, then \(\text{gr}({\mathcal H})=1\).