A combination theorem for special cube complexes.
type: Item
instance of: scholarly article
title: A combination theorem for special cube complexes.
zbMATH Open document ID: 1277.20046
author: Frédéric Haglund
author: Daniel T. Wise
published in: Annals of Mathematics. Second Series
publication date: 2013-01-03T00:00:00Z
review text: The paper under review gives a high-dimensional generalization of the one-dimensional work of \textit{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 [\textit{D. T. Wise}, op. cit.] has a natural extension to the ``foldable complexes'' of \textit{W. Ballmann} and \textit{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 d…
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 6121646
zbMATH Keywords: CAT(0) cube complexes
zbMATH Keywords: graphs of groups
zbMATH Keywords: residual finiteness
zbMATH Keywords: right-angled Artin groups
zbMATH Keywords: virtually special complexes
zbMATH Keywords: word-hyperbolic groups
reviewed by: Gregory C. Bell
MaRDI profile type: Publication
full work available at URL: annals.2012.176.3.2
OpenAlex ID: W2126761065
Wikidata QID: Q56519834
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DOI: 10.4007/ANNALS.2012.176.3.2
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description: scientific article; zbMATH DE number 6121646