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Anton A. Klyachko

Publications and source records attributed to Anton A. Klyachko.

41 records · Page 3Linked to original sources
arXiv2006

How to generalize known results on equations over groups

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular equations over torsion free groups, we prove an analogue of Magnus's Freiheitssatz, which asserts that there exists a solution with good behavior with respect to free factors of the initial group.

math.GR↗
arXiv2006

The SQ-universality of one-relator relative presentations

Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance, adding one generator and one relator in which the exponent sum of the additional generator is one to a free product of two nontrivial torsion-free groups, we also obtain an SQ-universal group. Key words: relative presentations, one-relator groups, SQ-universality, equations over groups.

math.GR↗
arXiv2006

Free subgroups of one-relator relative presentations

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group G= always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.

math.GR↗
arXiv2004

The Kervaire-Laudenbach conjecture and presentations of simple groups

The statement ``no nonabelian simple group can be obtained from a nonsimple group by adding one generator and one relator" 1) is equivalent to the Kervaire--Laudenbach conjecture; 2) becomes true under the additional assumption that the initial nonsimple group is either finite or torsion-free. Key words: Kervaire--Laudenbach conjecture, relative presentations, simple groups, car motion, cocar comotion. AMS MSC: 20E32, 20F05, 20F06.

math.GR↗