| 研究実績の概要 |
A variety of groups is a class of groups which is closed with respect to subgroups, direct products and homomorphic images. Given a variety V of groups and a subset X of a group G, we say that G is V-free over X if G belongs to V, and every map f from X to a group H from the variety V admits a unique extension to a homomorphism from G to H. A group G is V-free if it is V-free over some of its subsets.
We prove that Markov and Zariski topologies coincide for V-free groups, for every variety V of groups, thereby solving 79 years old problem of Markov for V-free groups. When V is the variety of all groups, this implies that all free groups have coinciding Markov and Zariski topologies. This particular case was obtained earlier by the author and Victor Hugo Yanez. The key to the proof of main result is the following theorem. For every countable subset Y of a set X, every Hausdorff group topology on the V-free group with alphabet Y can be extended to a Hausdorff group topology on the V-free group with alphabet X. (Here V is an arbitrary variety of groups.)
We expect that new technique developed for proving these results would help to find a characterization of countable Zariski dense sets in free (and more generally, V-free) groups.
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