Volume 18, issue 1 (2018)

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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

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 × H and G H have asymptotic property C. We also prove that a group G has asymptotic property C if 1 K G H 1 is exact, asdimK < 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.

asymptotic property C, asymptotic dimension, coarse geometry
Mathematical Subject Classification 2010
Primary: 54F45
Secondary: 20F69
Received: 22 July 2016
Revised: 29 June 2017
Accepted: 21 July 2017
Published: 10 January 2018
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