On the stability of asymptotic property C for products and some group extensions

Bell, Gregory; Nagórko, Andrzej

doi:10.2140/agt.2018.18.221

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On the stability of asymptotic property C for products and some group extensions

Gregory Copeland Bell and Andrzej Nagórko

Algebraic & Geometric Topology 18 (2018) 221–245

DOI: 10.2140/agt.2018.18.221

Abstract

We show that Dranishnikov’s asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if $G$ and $H$ are groups with asymptotic property C, then both $G \times H$ and $G * H$ have asymptotic property C. We also prove that a group $G$ has asymptotic property C if $1 \to K \to G \to H \to 1$ is exact, $asdim K < \infty$ and $H$ has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk (2016), of Bell and Moran (2015) and an open problem in topology.

Keywords

asymptotic property C, asymptotic dimension, coarse geometry

Mathematical Subject Classification 2010

Primary: 54F45

Secondary: 20F69

References

Publication

Received: 22 July 2016

Revised: 29 June 2017

Accepted: 21 July 2017

Published: 10 January 2018

Authors

Gregory Copeland Bell
Department of Mathematics and Statistics The University of North Carolina Greensboro, NC United States

Andrzej Nagórko
Faculty of Mathematics, Informatics, and Mechanics University of Warsaw Warszawa Poland

Volume 18, issue 1 (2018)