On \(p\)-deficiency in groups. (Q2869860)

In a recent paper [J. Group Theory 15, No. 2, 261-270 (2012; Zbl 1259.20046)] the second author developed remarkably simple arguments to produce new examples of finitely generated infinite \(p\)-groups. The main tool in his construction is the \(p\)-deficiency \(\text{def}_p(G)\) of a finitely generated group \(G\). Suppose that \(\langle X\mid R\rangle\) is a presentation for \(G\) with finite generating set \(X\). The \(p\)-valuation \(\nu_p(r)\) of a relator \(r\in R\) is the largest integer \(k\) such that \(r\) can be written as a \(p^k\)-th power of an element of the free group \(F(X)\) on \(X\). The \(p\)-deficiency of the presentation \(\langle X\mid R\rangle\) is \(|X|-1-\sum_{r\in R}p^{-\nu_p(r)}\) and the \(p\)-deficiency \(\text{def}_p(G)\) of \(G\) is the supremum taken over all presentations of \(G\) with a finite generating set.NEWLINENEWLINE A crucial role in that paper was played by the super-multiplicity of the \(p\)-deficiency for normal subgroup of index \(p\)-power, which allows to prove that if \(G\) has non-negative \(p\)-deficiency, then \(\Gamma\) is infinite and has an infinite pro-\(p\) completion. Usually, knowing that the pro-\(p\) completion of a group is non-trivial tells us nothing about the pro-\(p\) completion of a subgroup of finite index, unless the index is a \(p\)-power. Quite surprisingly, the authors obtain in this paper a generalization of the super-multiplicity to all normal subgroups of finite index: let \(G\) be a finitely generated group and let \(H\) be a normal subgroup of finite index in \(G\). Then \(\text{def}_p(H)\geq |G:H|\text{def}_p(H)\). In particular if \(\text{def}_p(G)\) is non-negative (resp. positive), then \(\text{def}_p(H)\) is non-negative (resp. positive); furthermore \(H\) has a non-trivial pro-\(p\) completion. The authors use the methods of the proof to show that some groups with non-positive \(p\)-deficiency have virtually positive \(p\)-deficiency. They also compute the \(p\)-deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute \(p\)-homology gradients and the \(p\)-Euler characteristic.