** "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.