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Properly \(3\)-realizable groups. (English) Zbl 1062.57001
A finitely presented group \(G\) is properly 3-realizable if there exists a compact 2-polyhedron \(K\) with fundamental group \(G\) whose universal cover has the proper homotopy type of a 3-manifold with boundary.
The authors show that if \(H\) and \(G\) are finitely presented groups where \(H\) is a subgroup of finite index of \(G\), then \(G\) is properly 3-realizable if and only if \(H\) is.
They also show that if \(G\) is properly 3-realizable and \(K\) is a 2-polyhedron with fundamental group \(G\) then the universal cover of \(K\vee S^2\) has the proper homotopy type of a 3-manifold.
Their last result is that the fundamental group of a finite graph of groups with properly 3-realizable vertex groups and finite cyclic edge groups is properly 3-realizable.

57M07 Topological methods in group theory
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M10 Covering spaces and low-dimensional topology
57M20 Two-dimensional complexes (manifolds) (MSC2010)
Full Text: DOI
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