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Infinitely presented small cancellation groups have the Haagerup property

Goulnara Arzhantseva

Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

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and

Damian Osajda

Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland

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https://doi.org/10.1142/S1793525315500144Cited by:3 (Source: Crossref)

Abstract

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the C'(1/6)-small cancellation condition. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum–Connes conjecture holds for them. The result is a first nontrivial advancement in understanding groups with such properties among infinitely presented non-amenable direct limits of hyperbolic groups. The proof uses the structure of a space with walls introduced by Wise. As the main step we show that C'(1/6)-complexes satisfy the linear separation property.

In Memory of Kamil Duszenko

Keywords:

AMSC: 20F06, 20F67, 46B85, 46L80

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