"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.