Advertisement

Lobachevskii Journal of Mathematics

, Volume 39, Issue 2, pp 218–223 | Cite as

Nontrivial Pseudocharacters on Groups and Their Applications

  • D. Z. Kagan
Article
  • 10 Downloads

Abstract

In this paper questions of the existence of non-trivial pseudocharacters for different classes of groups and the most important applications of pseudocharacters are considered. We review the results obtained for non-trivial pseudocharacters of free group constructions. Pseudocharacter is the real functions f on a group G such that 1) |f(ab) − f(a) − f(b)| ≤ ε for some ε > 0 and for any a, bG and 2) f(x n ) = nf(x) ∀n ∈ Z, ∀xG. Existence of non-trivial pseudocharacters implies the results for second group of bounded cohomologies and the width of verbal subgroups. Results of R.I. Grigorchuk, V.G. Bardakov, V.A. Fayziev and author on this topic are examined. Theorems about conditions of the existence of non-trivial pseudocharacters on such group objects as free products with amalgamation, HNN-extensions, group with one defining relation, anomalous products are given in the article.

Keywords and phrases

non-trivial pseudocharacter bounded cohomology width of verbal subgroups groups with one defining relation locally indicable groups anomalous products HNN-extensions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Gromov, “Volume and bounded cohomology,” Inst. Hautes Etudes Sci. Publ.Math. 56, 5–99 (1982).MathSciNetzbMATHGoogle Scholar
  2. 2.
    B. E. Johnson, “Cohomology in Banach algebras,” Mem. Am.Math. Soc. 127 (1972).Google Scholar
  3. 3.
    Y. I. Merzlyakov, Rational Groups (Nauka, Moscow, 1987) [in Russian].zbMATHGoogle Scholar
  4. 4.
    A. H. Rhemtulla, “A problem of bounded expressibility in free products,” Proc. Cambridge Phil. Soc. 64, 573–584 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. I. Grigorchuk, “Bounded cohomology of group constructions,” Mat.Zam. 59, 546–550 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. I. Grigorchuk, “Some results on bounded cohomology,” in Combinatorial and Geometric Group Theory, Vol. 204 of London Mathematical Society Lecture Note Series (London Math. Soc., Edinburgh, 1993), pp. 111–163.Google Scholar
  7. 7.
    V. G. Bardakov, “On a width of verbal subgroups of certain free constructions,” Algebra Logic 36, 494–517 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    I. V. Dobrynina and D. Z. Kagan, “On a width of verbal subgroups of certain free constructions,” Chebyshev. Sb. 16 (4), 150–163 (2015).Google Scholar
  9. 9.
    A. I. Shtern, “Quasirepresentations and pseudorepresentations,” Function Anal. Appl. 25 (2), 70–73 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. A. Faiziev, “The stability of a functional equation on groups,” Usp. Mat. Nauk 48, 193–194 (1993).MathSciNetGoogle Scholar
  11. 11.
    V. A. Faiziev, “Pseudocharacters on free groups and on certain group constructions,” Russ. Math. Surv. 43, 219–220 (1988).CrossRefGoogle Scholar
  12. 12.
    I. V. Dobrynina, “Solution of the problem of expressibility in amalgamated products of groups,” Fundam. Prikl. Mat. 15 (1), 23–30 (2009).Google Scholar
  13. 13.
    I. V. Dobrynina and V. N. Bezverkhnii, “On width in some class of groups with two generators and one defining relation,” Proc. Steklov Inst.Math. Algebra Topol. 7 (2), 53–60 (2001).MathSciNetzbMATHGoogle Scholar
  14. 14.
    D. Z. Kagan, “About exist non-trivial pseudocharacters on the anomalous product of groups,” Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 6, 24–28 (2004).Google Scholar
  15. 15.
    D. Z. Kagan, “Pseudocharacters on anomalous products of locally indicable groups,” Fundam. Prikl. Mat. 12 (3), 55–64 (2006).MathSciNetGoogle Scholar
  16. 16.
    D. Z. Kagan, “Nontrivial pseudocharacters on groups with one defining relation and nontrivial centre,” Sb.: Math. 208 (1), 75–89 (2017).MathSciNetzbMATHGoogle Scholar
  17. 17.
    D. Z. Kagan, “Pseudocharacters on free groups, invariant with respect to some types of endomorphisms,” Fundam. Appl.Math. 17, 167–176 (2011).Google Scholar
  18. 18.
    S. D. Brodskii, “Equations over groups and groups with a single defining relation,” Sib. Math. J. 25 (2), 84–103 (1980).MathSciNetGoogle Scholar
  19. 19.
    W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations, Dover Books onMathematics (Dover, New York, 2004).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Peoples Friendship University of Russia (RUDN)MoscowRussia

Personalised recommendations