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Resolutions of CAT(0) cube complexes and accessibility properties

Benjamin Beeker and Nir Lazarovich

Algebraic & Geometric Topology 16 (2016) 2045–2065
Abstract

In 1985, Dunwoody defined resolutions for finitely presented group actions on simplicial trees, that is, an action of the group on a tree with smaller edge and vertex stabilizers. Moreover, he proved that the size of the resolution is bounded by a constant depending only on the group. Extending Dunwoody’s definition of patterns, we construct resolutions for group actions on a general finite-dimensional CAT(0) cube complex. In dimension two, we bound the number of hyperplanes of this resolution. We apply this result for surfaces and 3–manifolds to bound collections of codimension-1 submanifolds.

Keywords
geometric group theory, CAT(0) cube complexes, 3–manifolds , actions on trees
Mathematical Subject Classification 2010
Primary: 20E08
Secondary: 20F65
References
Publication
Received: 26 February 2015
Revised: 11 June 2015
Accepted: 12 September 2015
Published: 12 September 2016
Authors
Benjamin Beeker
Department of Mathematics
Hebrew University
9190401 Jerusalem
Israel
Nir Lazarovich
Department of Mathematics
ETH Zürich
Rämistrasse 101
CH-8092 Zürich
Switzerland