Fully residually free pro-\(p\) groups.
The aim of the paper under review is to initiate the study of fully residually free pro-\(p\) groups. A pro-\(p\) group \(G\) is said to be fully residually free pro-\(p\) if for every finite subset \(X\subseteq G\) there exists a continuous homomorphism \(G\to F\) to a free pro-\(p\) group \(F\) which is injective on~\(X\). One says that \(G\) is residually free pro-\(p\) if the above condition holds for \(|X|=2\). The study of abstract (fully) residually free groups goes back to \textit{G.~Baumslag} [Math.\ Z.\ 78, 423-438 (1962; Zbl 0104.24402), Proc.\ Lond.\ Math.\ Soc., III. Ser. 17, 402-418 (1967; Zbl 0166.01502)]. More recently it was shown that, in the setting of abstract groups, the class of finitely generated, fully residually free groups coincides with the class of so-called limit groups. The latter have important applications, for instance, in Sela's solution of Tarski's problem about equivalence of first order theories of finitely generated non-Abelian free groups. A class of pro-\(p\) groups which can be regarded as analogues of limit groups via extensions of centralisers was considered by \textit{D.~H.~Kochloukova} and \textit{P.~A.~Zalesskii} in a separate paper [Math.\ Z.\ 267, No. 1-2, 109-128 (2011; Zbl 1216.20024)]. The core contribution of the paper under review is a technical theorem which embeds a particular type of amalgamated free pro-\(p\) product into a direct product of free pro-\(p\) groups, in analogy to a result of G.~Baumslag [loc.\ cit.] for abstract groups. The proof uses the pro-\(p\) version of the Kurosh Subgroup Theorem, Fox calculus and certain relation modules. The authors then deduce that every orientable surface pro-\(p\) group \[ G_d=\langle x_1,\dots,x_{2d}\mid [x_1,x_2]\cdots [x_{2d-1},x_{2d}]=1\rangle \] of even genus \(d\) is residually free pro-\(p\). In the special case \(p=2\), they can extend their result also to odd genera, by passing to suitable open subgroups. Subsequently it is observed that a collection of equivalent characterisations of abstract fully residually free groups, given by B.~Baumslag [loc.~cit.], holds analogously in the pro-\(p\) setting. In particular, a residually free pro-\(p\) group \(G\) is fully residually free pro-\(p\) if and only if \(G\) is commutative transitive, i.e.\ if every non-trivial element of \(G\) has Abelian centraliser in~\(G\). As a consequence, it is noted that every orientable surface pro-\(p\) group \(G_d\) is fully residually free pro-\(p\), provided that \(d\) is even or \(p=2\). In a similar way it is shown that the free pro-\(p\) product of a free pro-\(p\) group of finite rank and a free Abelian pro-\(p\) group of rank \(2\) is fully residually free pro-\(p\).
scientific article; zbMATH DE number 5787073
scientific article