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 $ H_1,\dots, H_n$ of a free group $ F$ then the set $ H_1\cdots H_n$ is closed in the profinite topology of $ F$. This was the last and most difficult cornerstone to a proof of the long standing ``Type II conjecture'' on finite monoids (by J. Rhodes). Meanwhile there are two other proofs of the product theorem: one, based on [1], is using geometric methods and automata, another is by means of model theory [3]. Ribes and Zalesskii [5] also proved that ``profinite topology'' may be replaced with ``pro-$ \bf H$ topology'' where $ \bf H$ is any extension closed variety of finite groups (under the assumption that the groups $ H_i$ are pro-$ \bf H$ closed). This theorem has again interesting consequences to monoid theory, language theory, etc. While the assumption on $ \bf H$ 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 $ \bf H$ satisfying the much weaker condition: for each $ G\in \bf H$ there is a cyclic group $ C$ such that $ CwrG\in\bf H$ (here $ wr$ is the wreath product). The methods are geometric-combinatorial rather than algebraic. An essential ingredient is the (profinite) Cayley graph of the pro-$ \bf H$ completion $ \widehat{F_{\bf H}}$ of $ F$. This graph in general is not a profinite tree (not even a pro-$ p$ 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-$ p$ topology of a free group and algorithmic problems in semigroups, Internat. J. Algebra Comput. 4 (1994) 359-375.

E-Mail: karl.auinger@univie.ac.at

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