Asymptotic properties of groups acting on complexes

Author:
Gregory C. Bell

Journal:
Proc. Amer. Math. Soc. **133** (2005), 387-396

MSC (2000):
Primary 20F69; Secondary 20E08, 20E06

DOI:
https://doi.org/10.1090/S0002-9939-04-07630-0

Published electronically:
September 8, 2004

MathSciNet review:
2093059

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Abstract: We examine asymptotic dimension and property A for groups acting on complexes. In particular, we prove that the fundamental group of a finite, developable complex of groups will have finite asymptotic dimension provided the geometric realization of the development has finite asymptotic dimension and the vertex groups are finitely generated and have finite asymptotic dimension. We also prove that property A is preserved by this construction provided the geometric realization of the development has finite asymptotic dimension and the vertex groups all have property A. These results naturally extend the corresponding results on preservation of these large-scale properties for fundamental groups of graphs of groups. We also use an example to show that the requirement that the development have finite asymptotic dimension cannot be relaxed.

**1.**G. Bell,*Property A for groups acting on metric spaces*, Topology Appl.,130, No. 3, 2003, pp. 239-251. MR**1978888 (2004d:20047)****2.**G. Bell and A. Dranishnikov,*On asymptotic dimension of groups*, Algebr. Geom. Topol. 1, 2001, pp.57-71. MR**1808331 (2001m:20062)****3.**G. Bell and A. Dranishnikov,*On asymptotic dimension of groups acting on trees*(Submitted), 2002.**4.**M. Bridson and A. Haefliger,*Metric spaces of non-positive curvature*, Springer-Verlag, 1999. MR**1744486 (2000k:53038)****5.**A. Dranishnikov,*Asymptotic topology*, Russian Math. Surveys, 55, 2000, No 6, pp.71-116. MR**1840358 (2002j:55002)****6.**A. Dranishnikov and T. Januszkiewicz,*Every Coxeter group acts amenably on a compact space*, Topology Proc. 24, 1999, pp.135-141. MR**1802681 (2001k:20082)****7.**A. Dranishnikov, G. Gong, V. Lafforgue, and G. Yu,*Uniform embeddings into Hilbert space and a question of Gromov*, Canad. Math. Bull. 45, 2002, No. 1, pp.60-70. MR**1884134 (2003a:57043)****8.**S. Ferry, A. Ranicki, and J. Rosenberg,*A history and survey of the Novikov conjecture*in*Novikov conjectures, index theorems and rigidity*, vol. 1, S. Ferry, A. Ranicki, and J. Rosenberg, eds., Cambridge University Press, 1996. MR**1388295 (97f:57036)****9.**M. Gromov,*Asymptotic invariants of infinite groups*, in*Geometric Group Theory*, vol. 2, G. Niblo and M. Roller, eds., Cambridge University Press, 1993. MR**1253543 (94f:20002)****10.**M. Gromov,*Spaces and questions*, Geom. Funct. Anal., Special Volume, Part I, 2000, pp.118-161. MR**1826251 (2002e:53056)****11.**A. Haefliger,*Complexes of groups and orbihedra*, Group Theory from a Geometrical Viewpoint (E. Ghys, A. Haefliger, A. Verjovsky, eds.), Proc. ICTP Trieste 1990, World Scientific, Singapore, 1991, pp.504-540. MR**1170362 (93a:20001)****12.**N. Higson and J. Roe,*Amenable group actions and the Novikov conjecture*J. Reine Angew. Math. 519, 2000, pp.143-153. MR**1739727 (2001h:57043)****13.**P. Ostrand,*Covering dimension in general spaces*, General Topology Appl. 1, 1971, pp.209-221. MR**0288741 (44:5937)****14.**J.-P. Serre,*Trees*, Springer-Verlag, 1980, Translation of ``Arbres, Amalgames et ", Asterisque, 1977.**15.**J.-L. Tu,*Remarks on Yu's property A for discrete metric spaces and groups*, Bull. Soc. Math. France 129, 2001, pp.115-139. MR**1871980 (2002j:58038)****16.**G. Yu,*The Novikov conjecture for groups with finite asymptotic dimension*, Ann. of Math. 147, no. 2, 1998, pp.325-355. MR**1626745 (99k:57072)****17.**G. Yu,*The coarse Baum-Connes conjecture for groups which admit a uniform embedding into Hilbert space*, Inv. Math. 139, 2000, pp.201-240. MR**1728880 (2000j:19005)**

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Additional Information

**Gregory C. Bell**

Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
bell@math.psu.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07630-0

Keywords:
Complexes of groups,
asymptotic dimension,
property A

Received by editor(s):
December 5, 2002

Received by editor(s) in revised form:
September 23, 2003

Published electronically:
September 8, 2004

Communicated by:
Stephen D. Smith

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.