Resolutions of CAT(0) cube complexes and accessibility properties

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.

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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

Volume 16, issue 4 (2016)

Benjamin Beeker and Nir Lazarovich

Abstract