# Seminar Talk

### Characterizing virtually free pro-p groups that are topologically finitely generated

This is joint work with P.A. Zalesskii.
##### Abstract

The following statements about a topologically finitely generated pro-p group G are equivalent:
- G has a free pro-p subgroup of p-power index (is ``virtually free pro-p'')
- G is the pro-p completion of the extension of a free group by a finite p-group
- G is the pro-p fundamental group of a finite graph of finite p-groups

When G is free by cyclic of order p the result is due to Claus Scheiderer (1999).
The ingredients of the proof of our result comprise the pro-p analog of Bass-Serre theory due to Mel'nikov-Zalesskii, Scheiderer's result, and,
a deep result on p-adic representation theory by Alfred Weiss.
# Profinite Groups

Basic material on profinite groups will be covered. I am indebted to my students Sema Alıcı and
Berna Çınar for the Turkish
translation.
### Course Notes

are here buradadır
### Tentative schedule

- Tue, 31.1.2012 Basic Definitions (Projective limit, examples like p-adic numbers, infinite Galois theory)
- Thu, 2.2.2012 Sylow theory of profinite groups (and, in the same vein, Hall theory of prosolvable groups)
- Fr, 3.2.2012 Profinite topology and profinite completion (and, slightly more general, pro-C completions)
- Mo, 6.2.2012 Free pro-C groups and the pro-C Kurosh subgroup theorem
- Tue, 7.2.2012 Free constructions, amalgamated free products and HNN-extensions
- We, 8.2.2012 Profinite graphs (finite graphs only) Fundamental pro-C groups of finite graphs of finite groups