## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Sektion 1 - Algebra**

Dienstag, 18. September 2001, 16.00, Hörsaal 21

**Profinite topologies on the free group**
**Karl Auinger**, **Universität Wien**
(Koautor: Benjamin Steinberg)

The Ribes and Zalesskii product theorem [4] states that, given finitely
generated subgroups
of a free group then the set
is closed in the profinite topology of . This was the last and most
difficult cornerstone to a proof of the long standing ``Type II conjecture'' on
finite monoids (by J. Rhodes). Meanwhile there are two other proofs of the product
theorem: one, based on [1], is using geometric methods and automata, another is by means
of model
theory [3]. Ribes and Zalesskii [5] also proved that ``profinite topology'' may
be replaced with ``pro- topology'' where is any *extension closed
variety of finite groups* (under the assumption that the groups
are pro- closed). This theorem has again interesting consequences to monoid
theory, language theory, etc. While the assumption on of being
extension closed cannot be removed from that proof, the validity of the theorem is
not limited to such varieties. Using new methods we have shown that the result holds
for all varieties satisfying the much weaker condition: for each
there is a cyclic group such that
(here is the wreath
product). The methods are geometric-combinatorial rather than algebraic. An essential
ingredient is the (profinite) Cayley graph of the *pro- completion*
of . This graph in general
is not a *profinite tree* (not even a *pro- tree*) in the homological sense
[2], though its geometry is reminiscing of a tree.
The structure of this graph is investigated by the use of *inverse monoids*.

[1] |
C.J. Ash, *Inevitable graphs: A proof of the type II conjecture and
some related decision procedures*, Internat. J. Algebra Comput. **1** (1991)
127-146. |

[2] |
D. Gildenhuys and L. Ribes, *Profinite groups and Boolean graphs*, J.
Pure Appl. Algebra **12** (1978) 21-47. |

[3] |
B. Herwig and D. Lascar, *Extending partial automorphisms and the
profinite topology on free groups*, Trans. Amer. Math. Soc. **352** (2000)
1985-2021. |

[4] |
L. Ribes and P. Zalesskii, *On the profinite topology on a free
group*, Bull. London Math. Soc. **25** (1993) 37-43. |

[5] |
L. Ribes and P. Zalesskii, *The pro- topology of a free group
and algorithmic problems in semigroups*, Internat. J. Algebra Comput. **4**
(1994) 359-375. |

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