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A combination theorem for special cube complexes. (English) Zbl 1277.20046
The 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)]).

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
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[1] I. Agol, D. D. Long, and A. W. Reid, ”The Bianchi groups are separable on geometrically finite subgroups,” Ann. of Math., vol. 153, iss. 3, pp. 599-621, 2001. · Zbl 1067.20067
[2] I. Agol, Untitled, 2006.
[3] W. Ballmann and J. Świcatkowski, ”On groups acting on nonpositively curved cubical complexes,” Enseign. Math., vol. 45, iss. 1-2, pp. 51-81, 1999. · Zbl 0989.20029
[4] N. Bergeron, F. Haglund, and D. T. Wise, ”Hyperplane sections in arithmetic hyperbolic manifolds,” J. Lond. Math. Soc., vol. 83, iss. 2, pp. 431-448, 2011. · Zbl 1236.57021
[5] M. Bonk and O. Schramm, ”Embeddings of Gromov hyperbolic spaces,” Geom. Funct. Anal., vol. 10, iss. 2, pp. 266-306, 2000. · Zbl 0972.53021
[6] S. Buyalo and V. Schroeder, ”Embedding of hyperbolic spaces in the product of trees,” Geom. Dedicata, vol. 113, pp. 75-93, 2005. · Zbl 1090.54029
[7] V. N. Gerasimov, ”Semi-splittings of groups and actions on cubings,” in Algebra, Geometry, Analysis and Mathematical Physics (Russian), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997, pp. 91-109, 190. · Zbl 0906.20025
[8] R. Gitik, ”Ping-pong on negatively curved groups,” J. Algebra, vol. 217, iss. 1, pp. 65-72, 1999. · Zbl 0936.20019
[9] R. Gitik, M. Mitra, E. Rips, and M. Sageev, ”Widths of subgroups,” Trans. Amer. Math. Soc., vol. 350, iss. 1, pp. 321-329, 1998. · Zbl 0897.20030
[10] M. Gromov, ”Hyperbolic groups,” in Essays in Group Theory, New York: Springer-Verlag, 1987, vol. 8, pp. 75-263. · Zbl 0634.20015
[11] F. Haglund, Aspects combinatoires de la théorie géométrique des groupes.
[12] F. Haglund, ”Finite index subgroups of graph products,” Geom. Dedicata, vol. 135, pp. 167-209, 2008. · Zbl 1155.53025
[13] F. Haglund and D. T. Wise, ”Special cube complexes,” Geom. Funct. Anal., vol. 17, iss. 5, pp. 1551-1620, 2008. · Zbl 1155.53025
[14] F. Haglund and D. T. Wise, ”Coxeter groups are virtually special,” Adv. Math., vol. 224, iss. 5, pp. 1890-1903, 2010. · Zbl 1195.53055
[15] C. G. Hruska and D. T. Wise, Finiteness properties of cubulated groups, 2010. · Zbl 1335.20043
[16] C. G. Hruska and D. T. Wise, ”Packing subgroups in relatively hyperbolic groups,” Geom. Topol., vol. 13, iss. 4, pp. 1945-1988, 2009. · Zbl 1188.20042
[17] A. Minasyan, ”Separable subsets of GFERF negatively curved groups,” J. Algebra, vol. 304, iss. 2, pp. 1090-1100, 2006. · Zbl 1175.20034
[18] M. A. Roller, Poc-sets, median algebras and group actions. An extended study of Dunwoody’s construction and Sageev’s theorem.
[19] M. Sageev, ”Ends of group pairs and non-positively curved cube complexes,” Proc. London Math. Soc., vol. 71, iss. 3, pp. 585-617, 1995. · Zbl 0861.20041
[20] P. Scott, ”Subgroups of surface groups are almost geometric,” J. London Math. Soc., vol. 17, iss. 3, pp. 555-565, 1978. · Zbl 0412.57006
[21] D. T. Wise, ”Subgroup separability of graphs of free groups with cyclic edge groups,” Q. J. Math., vol. 51, iss. 1, pp. 107-129, 2000. · Zbl 0991.05056
[22] D. T. Wise, ”The residual finiteness of negatively curved polygons of finite groups,” Invent. Math., vol. 149, iss. 3, pp. 579-617, 2002. · Zbl 1040.20024
[23] D. W. Morris, Introduction to Arithmetic Groups, 2012.
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