A 2-generated just-infinite profinite group which is not positively generated. (Q1885651)

Let \(G\) be a profinite group. For a positive integer \(k\), define \(P_G(k)\) as the measure in \(G^k\) of the set of those \(k\)-tuples \((g_1,\dots,g_k)\) that generate \(G\) (topologically). \(G\) is said to be positively finitely generated (PFG) if for some \(k\) one has \(P_G(k)>0\), that is, the probability that \(k\) random elements generate \(G\) is positive. Based among others on examples of \textit{W.~M.~Kantor} and \textit{A.~Lubotzky} [Geom. Dedicata 36, No. 1, 67-87 (1990; Zbl 0718.20011)], L.~Pyber has asked whether every finitely generated profinite group which is not PFG admits an infinite epimorphic image which is PFG. In the paper under review the author shows that the answer is negative, by constructing a \(2\)-generated profinite group which is not PFG, and all of whose proper epimorphic images are finite.