2015/01/05 by K. Auinger, Karl Auinger, Auinger, K. · 1 citation
Computer Science · Mathematics · #05C25 #20E18 #20F65 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:05C25 #msc:20E18 #msc:20F65 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1501.00839
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)
openalex publication_date 2015/01/05 · arxiv created 2016/01/21 · arxiv updated 2016/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a formation \mathfrakF of finite groups, tight connections are established between the pro-\mathfrakF-topology of a finitely generated free group F and the geometry of the Cayley graph Γ(F_\mathfrakF) of the pro-\mathfrakF-completion F_\mathfrak F of F. For example, the Ribes--Zalesskii-Theorem is proved for the pro-\mathfrakF-topology of F in case Γ(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.