×

Pseudocharacters on anomalous products of locally indicable groups. (English. Russian original) Zbl 1167.20304

J. Math. Sci., New York 149, No. 3, 1224-1229 (2008); translation from Fundam. Prikl. Mat. 12, No. 3, 55-64 (2006).
Summary: The question on the existence of nontrivial pseudocharacters on anomalous products of locally indicable groups is considered. Some generalizations of theorems of R. I. Grigorchuk and V. G. Bardakov on the existence of nontrivial pseudocharacters on free products with amalgamated subgroup are found. It is proved that they exist on an anomalous product \(\langle G,x\mid w=1\rangle\), where \(G\) is a locally indicable noncyclic group. We also prove some other propositions on the existence of nontrivial pseudocharacters on anomalous products of groups. Results on the second cohomologies of these products and their nonamenability follow from the propositions on the existence of nontrivial pseudocharacters on these groups.

MSC:

20C15 Ordinary representations and characters
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. G. Bardakov, ”On the width of verbal subgroups of certain free constructions,” Algebra Logika, 36, No. 5, 494–517 (1997). · Zbl 0941.20017
[2] S. D. Brodsky, ”Equations over groups, and groups with one defining relation,” Sib. Mat. Zh., 25, No. 2, 84–103 (1984). · Zbl 0574.41004
[3] M. M. Cohen and C. Rourke, ”The surjectivity problem for one-generator, one-relator extensions of torsion-free groups,” Geom. Topol., 5, 127–142 (2001). · Zbl 1014.20015
[4] V. A. Fajziev, ”The stability of a functional equation on groups,” Russ. Math. Surv., 48, No. 1, 165–166 (1993). · Zbl 0804.34075
[5] R. I. Grigorchuk, ”Some results on bounded cohomology,” in: A. J. Duncan (ed.) et al., Combinatorial and Geometric Group Theory. Proc. of a Workshop Held at Heriot-Watt University, Edinburgh, GB, Spring of 1993, London Math. Soc. Lect. Notes Ser., Vol. 284, Cambridge Univ. Press, Cambridge (1995), pp. 111–163. · Zbl 0853.20034
[6] R. I. Grigorchuk, ”Bounded cohomology of group constructions,” Mat. Zametki, 59, No. 4, 546–550 (1996). · Zbl 0873.20039
[7] D. Z. Kagan, ”Existence of nontrivial pseudocharacters on anomalous group products,” Moscow Univ. Math. Bull., 59, No. 6, 24–28 (2004). · Zbl 1084.20005
[8] A. A. Klyachko, The Kervaire-Laudenbach Conjecture and Presentations of Simple Groups, arXiv: math.GR/0409146 (2004).
[9] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer (1977).
[10] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Interscience (1966).
[11] A. I. Shtern, ”Quasirepresentations and pseudorepresentations,” Funct. Anal. Appl., 25, No. 2, 140–143 (1991). · Zbl 0737.22003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.