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Geometry of Infinitely Presented Small Cancellation Groups and Quasi-homomorphisms

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 January 2019

Goulnara Arzhantseva  and
Cornelia Druţu
Goulnara Arzhantseva
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria Email: goulnara.arzhantseva@univie.ac.at
Cornelia Druţu
Affiliation:
Mathematical Institute, AWB, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom Email: drutu@maths.ox.ac.uk
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Abstract

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We study the geometry of infinitely presented groups satisfying the small cancellation condition $C^{\prime }(1/8)$, and introduce a standard decomposition (called the criss-cross decomposition) for the elements of such groups. Our method yields a direct construction of a linearly independent set of power continuum in the kernel of the comparison map between the bounded and the usual group cohomology in degree 2, without the use of free subgroups and extensions.

Keywords

small cancellation theoryGreendlinger lemmaquasi-homomorphismbounded cohomology

MSC classification

Primary: 20F06: Cancellation theory; application of van Kampen diagrams
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 997 - 1018
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The first author was supported in part by the ERC grant ANALYTIC no. 259527, and by the Swiss NSF, under Sinergia grant CRSI22-130435. The second author was supported in part by the EPSRC grant no. EP/K032208/1 entitled “Geometric and analytic aspects of infinite groups”, by the project ANR Blanc ANR-10-BLAN 0116, acronym GGAA, and by the LABEX CEMPI.

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