Quasi-isometry invariance of group splittings. (English) Zbl 1129.20027

Papasoglu, Panos

Ann. Math. (2) 161, No. 2, 759-830 (2005).

The main result of this paper is the following Theorem: Let \(G\) be a one-ended, finitely presented group that is not commensurable to a surface group. Then, \(G\) splits over a two-ended group if and only if the Cayley graph of \(G\) is separated by a line.
Here, a quasi-line in the Cayley graph is a path-connected subspace \(L\) which, equipped with the length metric \(d_L\) induced from the metric \(d\) of the Cayley graph is quasi-isometric to \(\mathbb{R}\) and such that for any two sequences \((x_n)\) and \((y_n)\) in \(L\), we have \(d(x_n,y_n)\to\infty\) if \(d_L(x_n,y_n)\to\infty\).
An important consequence of the theorem is the following Corollary: Let \(G_1\) be a one-ended finitely presented group that is not commensurable to a surface group. If \(G_1\) splits over a two-ended group and \(G_2\) is quasi-isometric to \(G_1\) then \(G_2\) also splits over a two-ended group.
It is unknown whether the theorem is true for finitely generated groups without further assumptions. – The author’s proof uses the JSJ (Jaco-Shalen-Johannson) decomposition theory for finitely presented groups.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

20F65	Geometric group theory
20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M60	Group actions on manifolds and cell complexes in low dimensions
57M07	Topological methods in group theory
20F05	Generators, relations, and presentations of groups
20E34	General structure theorems for groups

Keywords:

group splittings; quasi-isometry invariants; JSJ decompositions; finitely generated groups; finitely presented groups; word-hyperbolic groups; Cayley graphs; numbers of ends