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 Mat. Sb., 2017, Volume 208, Number 1, Pages 80–96 (Mi msb8527)

Nontrivial pseudocharacters on groups with one defining relation and nontrivial centre

D. Z. Kagan

Moscow State University of Railway Communications

Abstract: The problem concerning existence conditions for nontrivial pseudocharacters on one-relator groups with nontrivial centre is completely solved. It is proved that a nontrivial pseudocharacter exists on a group of this type if and only if the group is nonamenable. A pseudocharacter is a real function on a group for which the set $\{f(xy)-f(x)-f(y); x, y\in F\}$ is bounded and $f( x^n)=nf(x)$ for all $n\in\mathbb{Z}$ and $x\in F$. The existence of pseudocharacters is related to many important characteristics and properties of groups, such as the cohomology groups and the width of verbal subgroups. From our results for pseudocharacters we obtain corollaries concerning the width of verbal subgroups and the second bounded cohomology group for the one-relator groups with nontrivial centre.
Bibliography: 21 titles.

Keywords: nontrivial pseudocharacters, one-relator groups, bounded cohomology, width of verbal subgroups, amenability.

DOI: https://doi.org/10.4213/sm8527

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English version:
Sbornik: Mathematics, 2017, 208:1, 75–89

Bibliographic databases:

UDC: 512.54
MSC: Primary 20C99; Secondary 20E05, 20E06

Citation: D. Z. Kagan, “Nontrivial pseudocharacters on groups with one defining relation and nontrivial centre”, Mat. Sb., 208:1 (2017), 80–96; Sb. Math., 208:1 (2017), 75–89

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8527
• https://doi.org/10.4213/sm8527
• http://mi.mathnet.ru/eng/msb/v208/i1/p80

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. Z. Kagan, “Invariantnye funktsii na svobodnykh gruppakh i spetsialnykh HNN-rasshireniyakh”, Chebyshevskii sb., 18:1 (2017), 109–122
2. D. Z. Kagan, “Nontrivial pseudocharacters on groups and their applications”, Lobachevskii J. Math., 39:2 (2018), 218–223
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