TAILIEUCHUNG - Đề tài "Quasi-isometry invariance of group splittings "

We show that a finitely presented one-ended group which is not commensurable to a surface group splits over a two-ended group if and only if its Cayley graph is separated by a quasi-line. This shows in particular that splittings over two-ended groups are preserved by quasi-isometries. 0. Introduction Stallings in [St1], [St2] shows that a finitely generated group splits over a finite group if and only if its Cayley graph has more than one end. | Annals of Mathematics Quasi-isometry invariance of group splittings By Panos Papasoglu Annals of Mathematics 161 2005 759 830 Quasi-isometry invariance of group splittings By Panos Papasoglu Abstract We show that a finitely presented one-ended group which is not commensurable to a surface group splits over a two-ended group if and only if its Cayley graph is separated by a quasi-line. This shows in particular that splittings over two-ended groups are preserved by quasi-isometries. 0. Introduction Stallings in St1 St2 shows that a finitely generated group splits over a finite group if and only if its Cayley graph has more than one end. This result shows that the property of having a decomposition over a finite group for a finitely generated group G admits a geometric characterization. In particular it is a property invariant by quasi-isometries. In this paper we show that one can characterize geometrically the property of admitting a splitting over a virtually infinite cyclic group for finitely presented groups. So this property is also invariant by quasi-isometries. The structure of group splittings over infinite cyclic groups was understood only recently by Rips and Sela R-S . They developed a JSJ-decomposition theory analog to the JSJ-theory for three manifolds that applies to all finitely presented groups. This structure theory underlies and inspires many of the geometric arguments in this paper. A different approach to the JSJ-theory for finitely presented groups has been given by Dunwoody and Sageev in D-Sa . Their approach has the advantage of applying also to splittings over Zn or even more generally over slender groups . Bowditch in a series of papers Bo 1 Bo 2 Bo 3 showed that a one-ended hyperbolic group that is not a triangle group splits over a two-ended group if and only if its Gromov boundary has local cut points. This characterization implies that the property of admitting such a splitting is invariant under quasi-isometries for hyperbolic groups. .

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