A combination theorem for special cube complexes.

*(English)*Zbl 1277.20046The paper under review gives a high-dimensional generalization of the one-dimensional work of D. T. Wise [in Invent. Math. 149, No. 3, 579-617 (2002; Zbl 1040.20024)]. There the result can be formulated as follows: the \(2\)-complex built by amalgamating two graphs along a malnormal immersed graph is “virtually special”.

The special cube complexes were introduced by the authors [in Geom. Funct. Anal. 17(2007), No. 5, 1551-1620 (2008; Zbl 1155.53025)] and are higher dimensional versions of graphs that admit canonical completion and retraction. The paper under review considers some of the more difficult properties of such spaces, whose simpler properties were studied by the authors [loc. cit.]. In particular, the authors show the following theorem: Let \(A\) and \(B\) be compact virtually special cube complexes with word-hyperbolic \(\pi_1\), and let \(A\leftarrow M\) and \(M\to B\) be local isometries of cube complexes such that \(\pi_1M\) is quasi-convex and malnormal in \(\pi_1A\) and \(\pi_1B\). Let \(X=A\cup_MB\) be the cube complex obtained by gluing \(A\) and \(B\) together with \(M\times [-1,1]\). Then \(X\) is special. This theorem should be widely applicable to combinatorial group theory, both because it is concerned with higher-dimensional groups and also because \(2\)-dimensional subgroups are not always fundamental groups of \(2\)-dimensional special cube complexes, but require higher dimensions.

The theorem from [D. T. Wise, op. cit.] has a natural extension to the “foldable complexes” of W. Ballmann and J. Świątkowski [Enseign. Math., II. Sér. 45, No. 1-2, 51-81 (1999; Zbl 0989.20029)]. In particular the authors show that if \(C\) is a negatively curved foldable cube complex then \(C\) has a finite cover \(\widehat C\) such that \(\widehat C\) is a special cube complex. Among other results, the authors prove that every word-hyperbolic group is quasi-isometric to a uniformly locally finite CAT(0) cube complex and deduce that such groups embed quasi-isometrically into a product of finitely many trees (first proved by S. Buyalo and V. Schroeder, [Geom. Dedicata 113, 75-93 (2005; Zbl 1090.54029)]).

The special cube complexes were introduced by the authors [in Geom. Funct. Anal. 17(2007), No. 5, 1551-1620 (2008; Zbl 1155.53025)] and are higher dimensional versions of graphs that admit canonical completion and retraction. The paper under review considers some of the more difficult properties of such spaces, whose simpler properties were studied by the authors [loc. cit.]. In particular, the authors show the following theorem: Let \(A\) and \(B\) be compact virtually special cube complexes with word-hyperbolic \(\pi_1\), and let \(A\leftarrow M\) and \(M\to B\) be local isometries of cube complexes such that \(\pi_1M\) is quasi-convex and malnormal in \(\pi_1A\) and \(\pi_1B\). Let \(X=A\cup_MB\) be the cube complex obtained by gluing \(A\) and \(B\) together with \(M\times [-1,1]\). Then \(X\) is special. This theorem should be widely applicable to combinatorial group theory, both because it is concerned with higher-dimensional groups and also because \(2\)-dimensional subgroups are not always fundamental groups of \(2\)-dimensional special cube complexes, but require higher dimensions.

The theorem from [D. T. Wise, op. cit.] has a natural extension to the “foldable complexes” of W. Ballmann and J. Świątkowski [Enseign. Math., II. Sér. 45, No. 1-2, 51-81 (1999; Zbl 0989.20029)]. In particular the authors show that if \(C\) is a negatively curved foldable cube complex then \(C\) has a finite cover \(\widehat C\) such that \(\widehat C\) is a special cube complex. Among other results, the authors prove that every word-hyperbolic group is quasi-isometric to a uniformly locally finite CAT(0) cube complex and deduce that such groups embed quasi-isometrically into a product of finitely many trees (first proved by S. Buyalo and V. Schroeder, [Geom. Dedicata 113, 75-93 (2005; Zbl 1090.54029)]).

Reviewer: Gregory C. Bell (Greensboro)

##### MSC:

20F65 | Geometric group theory |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

57M07 | Topological methods in group theory |

20F36 | Braid groups; Artin groups |

20F67 | Hyperbolic groups and nonpositively curved groups |

20E26 | Residual properties and generalizations; residually finite groups |

##### Keywords:

CAT(0) cube complexes; graphs of groups; residual finiteness; right-angled Artin groups; virtually special complexes; word-hyperbolic groups
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\textit{F. Haglund} and \textit{D. T. Wise}, Ann. Math. (2) 176, No. 3, 1427--1482 (2012; Zbl 1277.20046)

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##### References:

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