Geometry and Analysis on Groups Research Seminar

"Geometry and Analysis on Groups" Research Seminar

Time: 2015.11.03, 15:00--17:00
Location: Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock
Title: "Profinite graphs and the Ribes–Zalesskiĭ-Theorem."
Speaker: Karl Auinger (Universität Wien)
Abstract: The Ribes–Zalesskiĭ-Theorem asserts that the (setwise) product \(H_1\cdots H_n\) of any finite sequence \(H_1,\dots, H_n\) of finitely generated subgroups of a free group \(F\) is closed in the profinite topology of \(F\). I shall discuss the question for which varieties/formations \(\mathfrak{V}\) of finite groups this theorem is valid for the pro-\(\mathfrak{V}\)-topology of \(F\) (in the sense that \(H_1\cdots H_n\) is pro-\(\mathfrak{V}\)-closed provided that the groups \(H_i\) are pro-\(\mathfrak{V}\)-closed). It turns out that a certain geometric property (namely that of being tree-like) of the Cayley graph of the pro-\(\mathfrak{V}\)-completion \(\widehat{F_\mathfrak{V}}\) of \(F\) is essential for this behaviour and (almost) provides a full answer. I shall further discuss a necessary and sufficient condition on a profinite group to admit a tree-like Cayley graph.