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Parabolic isometries of CAT(0) spaces and CAT(0) dimensions. (English) Zbl 1073.20035

Let \(G\) be a discrete group acting properly by isometries on a CAT(0) space \(X\). The CAT(0)-dimension of \(G\), \(\text{CAT(0)-dim\,}G\), is the minimal (covering) dimension of such an \(X\). When no such \(X\) exists, \(\text{CAT(0)-dim\,}G=\infty\). The geometric dimension of \(G\), \(\text{geom\,dim\,}G\) is the minimal dimension of a \(K(G,1)\) complex. It is always the case that \(\text{geom\,dim\,}G\leq\text{CAT(0)-dim\,}G\) for torsion-free \(G\). In the article under review the authors investigate the problem of finding a torsion-free \(G\) with \(\text{geom\,dim\,}G<\text{CAT(0)-dim\,}G<\infty\). Imposing the additional condition that the action is by semi-simple isometries, N. Brady and J. Crisp [Geom. Dedicata 94, 185-214 (2002; Zbl 1070.20043)] and M. R. Bridson [Math. Res. Lett. 8, No. 4, 557-567 (2001; Zbl 0990.20026)] found CAT(0) groups where the geometric dimension and CAT(0)-dimension differed by \(1\).
In this paper the authors study group actions on CAT(0) spaces with a fixed point in the ideal boundary. In certain situations the authors can replace the condition that the action is by semi-simple isometries with the condition that \(X\) is proper. Combining this with the results of Brady-Crisp or Bridson they conclude that there is a finitely generated group \(G\) with geometric dimension \(2\) such that (1) \(G\) does not act properly on any proper CAT(0) space of dimension \(2\) by isometries; and (2) \(G\) acts properly and co-compactly on some proper CAT(0) space of dimension \(3\) by isometries. They also introduce a new class of examples of groups having gaps between geometric and CAT(0)-dimension with the condition that \(X\) be proper. However, these groups do not act properly, co-compactly on a CAT(0) space, i.e., they are not CAT(0) groups.

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
57M20 Two-dimensional complexes (manifolds) (MSC2010)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
57M60 Group actions on manifolds and cell complexes in low dimensions
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References:

[1] S Adams, W Ballmann, Amenable isometry groups of Hadamard spaces, Math. Ann. 312 (1998) 183 · Zbl 0913.53012
[2] W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag (1995) · Zbl 0834.53003
[3] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser (1985) · Zbl 0591.53001
[4] M Bestvina, G Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991) 469 · Zbl 0767.20014
[5] N Brady, J Crisp, Two-dimensional Artin groups with \(\mathrm{CAT}(0)\) dimension three (2002) 185 · Zbl 1070.20043
[6] T Brady, Complexes of nonpositive curvature for extensions of \(F_2\) by \(\mathbbZ\), Topology Appl. 63 (1995) 267 · Zbl 0830.20054
[7] M R Bridson, Length functions, curvature and the dimension of discrete groups, Math. Res. Lett. 8 (2001) 557 · Zbl 0990.20026
[8] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer (1999) · Zbl 0988.53001
[9] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) · Zbl 0584.20036
[10] D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, American Mathematical Society (2001) · Zbl 0981.51016
[11] R Charney, M W Davis, Finite \(K(\pi, 1)\)s for Artin groups, Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 110 · Zbl 0930.55006
[12] S S Chen, L Greenberg, Hyperbolic spaces, Academic Press (1974) 49 · Zbl 0295.53023
[13] S Deo, K Varadarajan, Discrete groups and discontinuous actions, Rocky Mountain J. Math. 27 (1997) 559 · Zbl 0897.57025
[14] K Fujiwara, K Nagano, T Shioya, Fixed point sets of parabolic isometries of CAT(0)-spaces, Comment. Math. Helv. 81 (2006) 305 · Zbl 1098.53025
[15] M Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 · Zbl 0841.20039
[16] M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser (2001) · Zbl 0958.57001
[17] N Monod, Superrigidity for irreducible lattices and geometric splitting, C. R. Math. Acad. Sci. Paris 340 (2005) 185 · Zbl 1057.22015
[18] J i Nagata, Modern dimension theory, Sigma Series in Pure Mathematics 2, Heldermann Verlag (1983) · Zbl 0518.54002
[19] A R Pears, Dimension theory of general spaces, Cambridge University Press (1975) · Zbl 0312.54001
[20] T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. (1987) · Zbl 0611.57002
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