On the width of verbal subgroups in some classes of groups

I. V. Dobrynina ; D. Z. Kagan

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 150-163.

Résumé

In this paper the problem of the width for verbal subgroups in different classes of groups is considered. We give a review the results obtained in this direction. The width of the verbal subgroups $V (G) $ is equal to а least value of $m\in \mathcal{N}\bigcup \{+\infty \}$ such that every element of the subgroup $ V (G) $ is represented as the product of at most $m$ values of words $V^{\pm 1}.$ The results about the width of verbal subgroups for free products and other free group constructions, such as free products with amalgamation and $HNN$-extensions are indicated. A. H. Rhemtulla solved the question of conditions for infinity of the width of any proper verbal subgroups in free products. V. G. Bardakov and I. V. Dobrynina received similar results for the free products with amalgamation and $HNN$-extensions, for which associated subgroups are different from the base group. Also, V. G. Bardakov completely solved the problem of the width of verbal subgroups in the group of braid. Many mathematicians studied the width of verbal subgroups generated by words from commutator subgroup for some classes of groups. R. I. Grigorchuk found conditions for infinity such verbal subgroups of free products with amalgamation and $HNN$-extensions, for which associated subgroups are \linebreak different from the base group. D. Z. Kagan obtained the corresponding results on width of verbal subgroups generated by words from commutator subgroup for groups with one defining relation and two generators, having a non-trivial center. Authors obtained the results about infinity of the width of verbal subgroups for groups with certain presentations, as well as for anomalous products of various types of groups. Also many results about verbal subgroups of Artin and Coxeter groups and graph groups are considered in the article. Bibliography: 25 titles.

Détail

Keywords: width of verbal subgroup, amalgamated free products, $HNN$-extensions.

Bibliographie

[1] Merzlyakov Y. I., Rational groups, Nauka, M., 1987 | MR

[2] Merzlyakov Y. I., “Algebraic linear groups as full groups of automorphisms and closure of their verbal subgroups”, Algebra and logic, 6:1 (1967), 83–94 | MR | Zbl

[3] Rhemtulla A. H., “A problem of bounded expressibility in free products”, Proc. Cambridge Phil. Soc., 64:3 (1968), 573–584 | DOI | MR | Zbl

[4] Rhemtulla A. H., “A problem of bounded expressibility in free products”, Proc. Cambridge Phil. Soc., 64:3 (1968), 573–584 | DOI | DOI | MR | MR | Zbl | Zbl

[5] Grigorchuk R. I., “Bounded cohomology of group constructions”, Mat. Zametki, 59:4 (1996), 546–550 | DOI | MR | MR | Zbl | Zbl

[6] Dobrynina I. V., “On the width in free products with amalgamation”, Mat. Zametki, 68:3 (2000), 353–359 | DOI | MR | Zbl

[7] Fa\v{i}ziev V. A., “A problem of expressibility in some amalgamated products of groups”, J. Austral. Math. Soc., 71 (2001), 105–115 | DOI | MR

[8] Dobrynina I. V., “Solution of the width problem in amalgamated free products”, Fundam. Prikl. Mat., 15:1 (2009), 23–30

[9] Dobrynina I. V., Bezverkhnii V. N., “On width in some class of groups with two generators and one defining relation”, Proc. Steklov Inst. Math. Algebra. Topology, 2001, 53–60 | MR | MR

[10] Kagan D. Z., “Width of verbal subgroups for groups with one defining relation”, Algebra, number theory and discrete geometry: Modern Problems and Application, XIII International Conference (Tula, 2015), 76–78

[11] Kagan D. Z., “Pseudocharacters on free groups, invariant with respect to some types of endomorphisms”, Fundam. Prikl. Mat., 17:2 (2012), 167–176 | Zbl

[12] Ito N. A., “A theorem of alternating group $A_n \ (n\geq5)$”, Math. Japon, 2:2 (1951), 59–60 | MR | Zbl

[13] Ore S., “Some remarks on commutators”, Proc. Amer. Math. Soc., 2 (1951), 307–314 | DOI | MR | Zbl

[14] Kagan D. Z., “On the existence of non-trivial Pseudocharacters on anomalous products of groups”, Vestnik MGU, 2004, no. 6, 24–28

[15] Kagan D. Z., “Pseudocharacters on anomalous products of locally indicable groups”, Fundam. Prikl. Mat., 12:3 (2006), 55–64 | MR | Zbl

[16] Gluhov M. M., Zubov A. Y., “On the lengths of the symmetric and alternating groups of substitutions in the different systems forming”, Mathematical questions of cybernetics, 1999, no. 8, 5–32 | MR

[17] Kourov's Writing-Book. Unsolved problems in group theory, 10th edition, Novosibirsk, 1986

[18] Repin N. N., “On the commutator equations in the groups $B_3$ and $B_4$”, Algorithmic problems in group theory and semigroups (Tula, 1986), 114–117

[19] Semenov Y. S., “On the commutators in braid groups”, The 10th All-Union symposium on group theory, Abstracts (Minsk, 1986), 207

[20] Durnev V. G., On the width of the commutator subgroup of the braid groups $B_3$ and $B_4$", Dep. v VINITI, No 4040-V87, 1987

[21] Durnev V. G., Shalashov V. K., “On the width of the commutator subgroup of the braid groups $B_3$ and $B_4$”, The 19th All-Union algebraic conference, Abstracts (Lviv, 1987), 89

[22] Bardakov V. G., “On the theory of braid groups”, Mat. Sb., 183:6 (1992), 3–42

[23] Bardakov V. G., “The width of verbal subgroups of certain Artin groups, group and metric properties of mappings”, The collection of papers dedicated to the memory of Y. I. Merzlyakov, Novosibirsk, 1995, 8–18

[24] Bezverkhnii V. N., Dobrynina I. V., “Solution of the finite width problem in Artin groups with two generators”, Chebyshevskii Sb., 3:1(3) (2002), 11–16 | MR

[25] Myasnikov A., Nikolaev A., “Verbal subgroups of hyperbolic groups have infinite width”, J. Lond. Math. Soc.-Second Ser. II, 90:2 (2014), 573–591 | DOI | MR | Zbl