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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)


A Higman embedding preserving asphericity

Author: Mark Sapir
Journal: J. Amer. Math. Soc. 27 (2014), 1-42
MSC (2010): Primary 20F65; Secondary 20F69, 20F38, 22F50
Published electronically: July 9, 2013
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Abstract: We prove that every finitely generated group with recursive
aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold $ M$ such that the fundamental group $ \pi _1(M)$ coarsely contains an expander. Thus it has infinite asymptotic dimension, is not coarsely embeddable into a Hilbert space, does not satisfy G. Yu's property A, and does not satisfy the Baum-Connes conjecture with coefficients. Closed aspherical manifolds with any of these properties were previously unknown.

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Additional Information

Mark Sapir
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

PII: S 0894-0347(2013)00776-6
Keywords: Aspherical group, Higman embedding, $S$-machine
Received by editor(s): April 26, 2011
Received by editor(s) in revised form: September 22, 2011, November 29, 2011, and April 29, 2013
Published electronically: July 9, 2013
Additional Notes: This research was supported in part by NSF grant DMS-0700811.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.