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

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.

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 |