Karl Auinger, Universität Wien (Koautor: Benjamin Steinberg)
The Ribes and Zalesskii product theorem  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 , is using geometric methods and automata, another is by means of model theory . Ribes and Zalesskii  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 , though its geometry is reminiscing of a tree. The structure of this graph is investigated by the use of inverse monoids.
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