The geometry of profinite graphs revisited

TL;DR

This paper explores the geometric properties of profinite graphs associated with free groups, establishing new results and simpler proofs for the Ribes--Zalesskii-Theorem in the context of pro--topologies.

Contribution

It provides geometric proofs linking pro--topologies of free groups to Cayley graph structures, and constructs new examples of profinite groups with tree-like Cayley graphs.

Findings

Proves Ribes--Zalesskii-Theorem for pro--topology when Cayley graph is a tree.

Establishes geometric proofs avoiding inverse monoids.

Constructs new profinite groups with tree-like Cayley graphs.

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 $\Gamma(\hat{F_{\mathfrak{F}}})$ of the pro-$\mathfrak{F}$-completion $\hat{F_{\mathfrak {F}}}$ of $F$. For example, the Ribes--Zalesskii-Theorem is proved for the pro-$\mathfrak{F}$-topology of $F$ in case $\Gamma(\hat{F_{\mathfrak F}})$ is a tree-like graph. All these results are established by purely geometric proofs, without the use of inverse monoids which were indispensable in earlier papers, thereby giving more direct and more transparent proofs. 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.