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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)].

MSC:

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
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[1] DOI: 10.1016/0022-4049(93)90085-8 · Zbl 0805.57001 · doi:10.1016/0022-4049(93)90085-8
[2] DOI: 10.1007/BF02392898 · Zbl 0911.57001 · doi:10.1007/BF02392898
[3] DOI: 10.1007/s002220000063 · Zbl 1017.20034 · doi:10.1007/s002220000063
[4] Farb R, J. Di{\currency}. Geom. 44 (3) pp 435– (1996)
[5] DOI: 10.1007/s002220050210 · Zbl 0937.22003 · doi:10.1007/s002220050210
[6] DOI: 10.1007/s002220050337 · Zbl 0931.20035 · doi:10.1007/s002220050337
[7] Kapovich B, J. Di{\currency}. Geom. 69 (2) pp 279– (2005)
[8] DOI: 10.1007/s002220050145 · Zbl 0866.20033 · doi:10.1007/s002220050145
[9] DOI: 10.4007/annals.2003.158.115 · Zbl 1038.20016 · doi:10.4007/annals.2003.158.115
[10] DOI: 10.1142/S0218196705002347 · Zbl 1085.20026 · doi:10.1142/S0218196705002347
[11] DOI: 10.4007/annals.2005.161.759 · Zbl 1129.20027 · doi:10.4007/annals.2005.161.759
[12] DOI: 10.1007/BF02392599 · Zbl 1029.11038 · doi:10.1007/BF02392599
[13] Scott T, LMS Lect. Note Ser. 36 pp 137– (1979)
[14] DOI: 10.2307/1970577 · Zbl 0238.20036 · doi:10.2307/1970577
[15] Wh K, GAFA 11 pp 1327– (2002)
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