zbMATH — the first resource for mathematics

Foldable cubical complexes of nonpositive curvature. (English) Zbl 1055.20035
The author uses the following terminology: A ‘cubical complex’ \(X\) is a CW-complex obtained by gluing unit Euclidean cubes along their faces via isometries, with the requirements that these cubes inject into \(X\) and the intersection of the images of two cubes is either empty or equals the image of a cube. A cubical complex is said to be ‘dimensionally homogeneous’ if there is an integer \(n\geq 1\) such that each cube of \(X\) is a face of some \(n\)-cube. A ‘folding’ of \(X\) is a combinatorial map \(f\colon X\to C\) onto an \(n\)-cube \(C\) such that the restriction of \(f\) to each cube is injective. The complex \(X\) is equipped with its natural length metric. An FCC is then a connected foldable cubical complex that is dimensionally homogeneous, geodesically complete and having nonpositive curvature in the sense of Alexandrov.
In this paper, the author proves the following Theorem 1: Let \(X\) be an FFC of dimension \(n\). Then \(X\) has the structure of a graph of spaces where all the vertex and edge spaces are \((n-1)\)-dimensional FFCs and the maps from edge spaces to vertex spaces are combinatorial immersions.
As an application, the author proves the following results: Theorem 2: Let \(X\) be a finite FFC of dimension 3 with universal cover \(\widetilde X\) and let \(\Gamma\) be its group of deck transformations. Then, (1) if \(\widetilde X\) has rank one, then \(\widetilde X\) contains a periodic rank one geodesic in the 1-skeleton; (2) if \(\widetilde X\) has higher rank, then \(\widetilde X\) is isometric to the product of two CAT(0) FCCs;
Theorem 3: Let \(X\) be a finite FCC. Then any subgroup of \(\pi_1(X)\) either contains a free group of rank two or is virtually free Abelian.
Theorem 2 is related to the ‘rank rigidity conjecture’ of Ballmann and others. Theorem 3 is due to W. Ballmann and J. Świątkowski, who in fact introduced the notion of foldable cubical complex [see Enseign. Math., II. Sér. 45, No. 1-2, 51-81 (1999; Zbl 0989.20029)]. The author gives here a new proof of Theorem 3 which is shorter than the one by Ballmann and Świątkowski.
20F65 Geometric group theory
20E08 Groups acting on trees
20F67 Hyperbolic groups and nonpositively curved groups
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
57M07 Topological methods in group theory
Full Text: DOI EMIS EuDML arXiv
[1] W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag (1995) · Zbl 0834.53003
[2] W Ballmann, M Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. (1995) · Zbl 0866.53029
[3] W Ballmann, M Brin, Rank rigidity of Euclidean polyhedra, Amer. J. Math. 122 (2000) 873 · Zbl 1041.53029
[4] W Ballmann, S Buyalo, Periodic rank one geodesics in Hadamard spaces, preprint (2002) · Zbl 1167.53034
[5] W Ballmann, J Świ\catkowski, On groups acting on nonpositively curved cubical complexes, Enseign. Math. \((2)\) 45 (1999) 51 · Zbl 0989.20029
[6] , Group theory from a geometrical viewpoint, World Scientific Publishing Co. (1991) · Zbl 0809.00017
[7] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, Springer (1999) · Zbl 0988.53001
[8] M R Bridson, D T Wise, \(\mathcal{VH}\) complexes, towers and subgroups of \(F{\times}F\), Math. Proc. Cambridge Philos. Soc. 126 (1999) 481 · Zbl 0942.20009
[9] R Charney, The Tits conjecture for locally reducible Artin groups, Internat. J. Algebra Comput. 10 (2000) 783 · Zbl 1012.20035
[10] M W Davis, Nonpositive curvature and reflection groups, North-Holland (2002) 373 · Zbl 0998.57002
[11] M Davis, T Januszkiewicz, R Scott, Nonpositive curvature of blow-ups, Selecta Math. \((\)N.S.\()\) 4 (1998) 491 · Zbl 0924.53033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.