The geometry of profinite graphs revisited
| Autor: | Karl Auinger |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Profinite group
Cayley graph Structure (category theory) Inverse Geometry Group Theory (math.GR) 20E18 20F65 05C25 Graph Free group FOS: Mathematics Discrete Mathematics and Combinatorics Geometry and Topology Finitely-generated abelian group Mathematics::Representation Theory Mathematics - Group Theory Mathematics |
| Popis: | 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. Comment: 4 figures (v1); proof of Prop. 4.1 and several other clarifications included (v2); minor inaccuracies removed, stylistic improvements implemented, polished version (v3); proof of Theorem 3.6 included, arguments at the end of section 2 improved (v4); Theorem 3.1 included, three open problems stated (v5) |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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