A \(p\)-group with positive rank gradient. (Q2903542)

Burnside's problem asks whether a finitely generated, torsion group is necessarily finite. \textit{E. S. Golod} gave a negative solution to it [in Am. Math. Soc., Translat., II. Ser. 48, 103-106 (1965); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 28, 273-276 (1964; Zbl 0215.39202)]. One may inquire how large a finitely generated torsion group can be. The paper under review shows that such groups can be quite large, by constructing, for each \(d\geq 2\), a \(d\)-generated \(p\)-group which behaves asymptotically very much like a \(d\)-generated free pro-\(p\)-group \(F_d^p\). In fact for each \(d\geq 2\) and \(\varepsilon>0\) there is a \(d\)-generated group \(\Gamma\) such that a subgroup of index \(p^n\) requires \((d-\varepsilon)p^n\) generators, and the subgroup growth of \(\Gamma\) is bounded from below by \(s_{p^n}(F_d^p)^{1-\varepsilon}\).NEWLINENEWLINE These groups have positive rank gradient. This answers in the positive a question of Nikolov, who had asked whether there is a group of positive rank gradient such that none of its subgroups of finite index maps onto an infinite cyclic group. Here the rank gradient of the finitely-generated, residually finite, infinite group \(\Gamma\) along \(\mathcal N\) is defined as the infimum over \(\mathcal N\) of \(\liminf(d(N_i)/|\Gamma:N_i|)\), where \(\mathcal N=(N_i)_{i=1}^\infty\) ranges over all sequences of normal subgroups of finite index with trivial intersections.