Arbitrary \(p\)-gradient values. (Q2869862)

The absolute rank gradient of a finitely generated group \(G\) is defined by NEWLINE\[NEWLINE\text{RG}(G)=\inf_{[G:H]<\infty}(d(H)-1)/[G:H],NEWLINE\]NEWLINE where \(d(H)\) denotes the minimum number of generators of \(H\) and the infimum is taken over all finite index subgroups \(H\) of \(G\). The following conjecture is still open: for every positive real number \(\alpha\) there exists a finitely generated group \(G\) such that \(\text{RG}(G)=\alpha\).NEWLINENEWLINE The author of this paper proves the analogous question for the \(p\)-gradient: for any prime number \(p\) and any positive real number \(\alpha\) there exists a finitely generated group \(G\) such that the rank gradient of the pro-\(p\) completion of \(G\) is equal to \(\alpha\). This construction is used to show that there exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable and residually-\(p\).