##
**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.

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.

Reviewer: Gregory C. Bell (Greensboro)

### 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 |

### Keywords:

CAT(0) spaces; parabolic isometries; Artin groups; Heisenberg group; geometric dimension; cohomological dimension; discrete groups; dimension gaps; finitely presented groups; fundamental groups; complex hyperbolic manifolds; universal covers; solvable Baumslag-Solitar groups
PDF
BibTeX
XML
Cite

\textit{K. Fujiwara} et al., Algebr. Geom. Topol. 4, 861--892 (2004; Zbl 1073.20035)

### 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 |

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.