The geometry of profinite graphs revisited

Abstract

For a formation $\mathfrak{F}$ of finite groups, tight connections are established between the pro-$\mathfrak{F}$-topology of a finitely generated free group $F$ and the geometry of the Cayley graph $\Gamma (\widehat{F_{\mathfrak{F}}})$ of the pro-$\mathfrak{F}$-completion $\widehat{F_{\mathfrak{F}}}$ of $F$. For example, the Ribes–Zalesskii theorem is proved for the pro-$\mathfrak{F}$-topology of $F$ in case $\Gamma (\widehat{F_{\mathfrak{F}}})$ is a tree-like graph. All these results are established by purely geometric proofs, more directly and more transparently than in earlier papers, without the use of inverse monoids. Due to the richer structure provided by formations (compared to varieties), new examples of (relatively free) profinite groups with tree-like Cayley graphs are constructed. Thus, new topologies on $F$ are found for which the Ribes–Zalesskii theorem holds.