Group splittings and asymptotic topology. (English) Zbl 1160.20034

A result of Stallings implies that, for groups with more than two ends, splittings over finite groups are preserved by quasi-isometries. In the present paper, asymptotic (or coarse) topology is used to generalize this to other types of splittings. More precisely, if a finitely generated group \(G\) is isomorphic to the fundamental group of a finite graph of groups such that all edge and vertex groups are coarse \(\text{PD}(n)\) (Poincaré duality) groups of dimension \(n\) (e.g. fundamental groups of aspherical manifolds), and if a group \(H\) is quasi-isometric to \(G\) then (excluded some trivial situations) also \(H\) splits over a group quasi-isometric to an edge group of \(G\); also, the same conclusion holds if all edge groups of \(G\) are smaller than the vertex groups in the sense that they are dominated by coarse \(\text{PD}(n-1)\)-spaces.
The approach is based on an asymptotic version of the Jordan separation theorem due to Schwartz and a stronger version given recently by M. Kapovich and B. Kleiner in the context of coarse \(\text{PD}(n)\)-spaces and groups [J. Differ. Geom. 69, No. 2, 279-352 (2005; Zbl 1086.57019)]. Independent proofs of some of the results of the present paper are given in a paper by L. Mosher, M. Sageev and K. Whyte [Ann. Math. (2) 158, No. 1, 115-164 (2003; Zbl 1038.20016)].


20F69 Asymptotic properties of groups
20F65 Geometric group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M07 Topological methods in group theory
57P10 Poincaré duality spaces
Full Text: DOI arXiv


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