Polynomial index growth groups (Q2709981)

A group \(G\) has polynomial index growth (\(\mathbf{PIG}\)) if there exists a constant \(s\) such that \(|G^*:G^{*n}|\leq n^s\) for every finite quotient \(G^*\) of \(G\) and all positive integers \(n\). The purpose of this paper is to investigate such groups.NEWLINENEWLINENEWLINEThe authors compare this condition with other finiteness conditions such as polynomial subgroup growth (\(\mathbf{PSG}\)) and being boundedly generated (\(\mathbf{BG}\)). One of the main results is to show that there exist finitely generated, residually finite \(\mathbf{PIG}\) groups that are neither \(\mathbf{BG}\) nor linear. This answers questions by \textit{V. P. Platonov, A. S. Rapinchuk} [Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 3, 483-508 (1992; Zbl 0785.20025)] and \textit{D. Segal} [Groups, St. Andrews 1985, Lond. Math. Soc. Lect. Note Ser. 121, 307-314, 315-319 (1986; Zbl 0605.20035, Zbl 0605.20036)], respectively. To prove this result the authors study the exponents of finite simple groups. They prove that there exists a universal constant \(R\) such that if \(L\) is a finite simple group of Lie type then \(|L|\leq\exp(L)^R\). This confirms, in a strengthened form, a conjecture of \textit{M. Vaughan-Lee} and \textit{E. I. Zel'manov} [J. Aust. Math. Soc., Ser. A 67, No. 2, 261-271 (1999; Zbl 0939.20044)].NEWLINENEWLINENEWLINEThe authors also consider the subgroup growth of \(\mathbf{PIG}\) groups and give a structure theorem for profinite \(\mathbf{PIG}\) groups.