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Covering theory for complexes of groups. (English) Zbl 1182.57003

Summary: We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and isometry of universal covers. We characterize faithful complexes of groups and prove a conjugacy theorem for groups acting freely on polyhedral complexes. We also define an equivalence relation on coverings of complexes of groups, which allows us to construct a bijection between such equivalence classes, and subgroups or overgroups of a fixed lattice \(\Gamma\) in the automorphism group of a locally finite polyhedral complex \(X\).

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)
20E08 Groups acting on trees
20E42 Groups with a \(BN\)-pair; buildings
57M07 Topological methods in group theory
57M10 Covering spaces and low-dimensional topology
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References:

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[9] S. Lim, A. Thomas, Counting overlattices for polyhedral complexes (submitted for publication); S. Lim, A. Thomas, Counting overlattices for polyhedral complexes (submitted for publication) · Zbl 1275.20026
[10] Thomas, A., Lattices acting on right-angled buildings, Algebr. Geom. Topol., 6, 1215-1238 (2006) · Zbl 1128.22002
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