Algebraic & Geometric Topology

Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

Abstract

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady–Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension $2$ which do not act properly on any proper $CAT(0)$ spaces of dimension $2$ by isometries, although such actions exist on $CAT(0)$ spaces of dimension $3$.

Another example is the fundamental group, $G$, of a complete, non-compact, complex hyperbolic manifold $M$ with finite volume, of complex dimension $n≥2$. The group $G$ is acting on the universal cover of $M$, which is isometric to $Hℂn$. It is a $CAT(−1)$ space of dimension $2n$. The geometric dimension of $G$ is $2n−1$. We show that $G$ does not act on any proper $CAT(0)$ space of dimension $2n−1$ properly by isometries.

We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag–Solitar groups.

Article information

Source
Algebr. Geom. Topol., Volume 4, Number 2 (2004), 861-892.

Dates
Revised: 30 July 2004
Accepted: 13 September 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882535

Digital Object Identifier
doi:10.2140/agt.2004.4.861

Mathematical Reviews number (MathSciNet)
MR2100684

Zentralblatt MATH identifier
1073.20035

Citation

Fujiwara, Koji; Shioya, Takashi; Yamagata, Saeko. Parabolic isometries of CAT(0) spaces and CAT(0) dimensions. Algebr. Geom. Topol. 4 (2004), no. 2, 861--892. doi:10.2140/agt.2004.4.861. https://projecteuclid.org/euclid.agt/1513882535

References

• Scott Adams, Werner Ballmann, Amenable isometry groups of Hadamard spaces. Math. Ann. 312 (1998), 183–195.
• Werner Ballmann, “Lectures on spaces of nonpositive curvature". DMV Seminar, 25. Birkhauser, 1995
• Werner Ballmann, Mikhael Gromov, Victor Schroeder, Manifolds of nonpositive curvature. Progress in Mathematics, 61. Birkhauser (1985).
• Mladen Bestvina, Geoffrey Mess, The boundary of negatively curved groups. J. Amer. Math. Soc. 4 (1991), no. 3, 469-481.
• Noel Brady, John Crisp, Two-Dimensional Artin Groups with CAT(0) Dimension Three, in “Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000)”, Geometriae Dedicata 94, No1 (2002), 185-214.
• Thomas Brady, Complexes of nonpositive curvature for extensions of $F\sb 2$ by $Z$. Topology Appl. 63 (1995), no. 3, 267–275.
• Martin R. Bridson, Length functions, curvature and the dimension of discrete groups. Math. Res. Lett. 8 (2001), no. 4, 557-567.
• Martin R. Bridson, Andre Haefliger, “Metric spaces of non-positive curvature". Grundlehren der Mathematischen Wissenschaften 319. Springer, 1999.
• Kenneth S. Brown, “Cohomology of groups". Graduate Texts in Mathematics, 87. Springer, 1982.
• Dmitri Burago, Yuri Burago, Sergei Ivanov, A course in metric geometry. Graduate Studies in Mathematics, 33. AMS, 2001.
• Ruth Charney, Michael W. Davis, Finite $K(\pi, 1)$s for Artin groups. in “Prospects in topology", 110-124, Ann. of Math. Stud. 138, Princeton Univ. Press, 1995.
• S.S.Chen, L.Greenberg, Hyperbolic spaces, in “Contributions to Analysis", Academic Press, 49-87, (1974).
• Satya Deo, K. Varadarajan, Discrete groups and discontinuous actions. Rocky Mountain J. Math. 27 (1997), no. 2, 559–583.
• Koji Fujiwara, Koichi Nagano, Takashi Shioya, Fixed point sets of parabolic isometries of CAT($0$)-spaces. preprint. 2004.
• M.Gromov, Asymptotic invariants of infinite groups. in “Geometric group theory, Vol. 2 (Sussex 1991)", 1–295, LMS Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, (1993).
• Michael Kapovich, Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhauser, 2001.
• N.Monod, Superrigidity for irreducible lattices and geometric splitting. preprint, Dec 2003.
• J. Nagata, “Modern dimension theory", revised edition, Sigma Series in Pure Mathematics, 2. Heldermann Verlag, 1983.
• A.R.Pears, Dimension theory of general spaces. Cambridge University Press, 1975.
• Tammo. tom Dieck, Transformation groups. de Gruyter Studies in Mathematics, 8. Walter de Gruyter, 1987.