44 citations · 271 across the 30 of their papers we have counts for
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Markov's problem for free groups
Dmitri Shakhmatov, Víctor Hugo Yañez
We prove that every unconditionally closed subset of a free group is algebraic, thereby answering affirmatively a 76 years old problem of Markov for free groups. In modern terminol…
Topological groups all continuous automorphisms of which are open
Vitalij Chatyrko, Dmitri Shakhmatov
A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topologic…
Automorphism groups of dense subgroups of R^n
Vitalij Chatyrko, Dmitri Shakhmatov
By an automorphism of a topological group G we mean an isomorphism of G onto itself which is also a homeomorphism. In this article, we study the automorphism group Aut(G) of a dens…
A Kronecker-Weyl theorem for subsets of abelian groups
Dikran Dikranjan, Dmitri Shakhmatov
Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2^c and a countable family F o…
The Markov-Zariski topology of an abelian group
Dikran Dikranjan, Dmitri Shakhmatov
According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of element…
Reflection principle characterizing groups in which unconditionally closed sets are algebraic
Dikran Dikranjan, Dmitri Shakhmatov
We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we p…