The Ribes-Zalesskii product theorem and Cayley graphs of profinite groups








K. Auinger
University of Vienna, Austria

B. Steinberg
Universidade do Porto, Portugal






Let $A$ be a finite alphabet. The Ribes-Zalesskii product theorem states that, for any positive integer $n$ and any finitely generated subgroups $H_1,\dots, H_n$ of the free group $F$ on $A$, the set $H_1\cdots H_n$ is closed in the profinite topology of $F$. Ribes and Zalesskii even proved that ``profinite topology'' may be replaced with ``pro-$\bf H$ topology'' where $\bf H$ is any extension closed pseudovariety of finite groups (under the assumption that the groups $H_i$ are pro-$\bf H$ closed). Recent research by B. Steinberg and myself shows that this theorem holds under a much weaker assumption on $\bf H$: namely that the (profinite) Cayley graph of the pro-$\bf H$ completion $\widehat{F_{\bf H}}$ of $F$ behaves geometrically like a tree. I shall discuss requirements on an inverse system $\{G_i\mid i\in I\}$ of $A$-generated finite groups $G_i$ in order that $\varprojlim G_i$ admits such a ``tree-like'' Cayley graph. It turns out that this happens when each $G_i$ admits a ``sufficiently good'' co-extension among the groups $G_j$ (this can be made precise). As a consequence, one gets that for a pseudovariety $\bf H$ of groups sufficient for the free pro-$\bf H$ group $\widehat{F_{\bf H}}$ to have a tree-like Cayley graph is that for each group $G\in \bf H$ there exists a cyclic group $C\ne 1$ for which the wreath product $C\wr G$ also belongs to $\bf H$; this is obviously much weaker than being extension closed.





Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.

 


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