×

On asymptotic dimension of groups. (English) Zbl 1008.20039

The authors extend Gromov’s notion of asymptotic dimension to answer questions relating to the finite asymptotic dimensionality of amalgamated products and HNN extensions. A theorem involving countable unions of metric spaces for asymptotic dimension is proved and the results are applied to groups acting on asymptotically finite dimensional metric spaces.
After proving results on the asymptotic dimension of finitely generated groups acting by isometries on metric spaces, the authors investigate sequences of groups with norms, \(\{A_i,\|\;\|_i\}\), to determine bounds on the asymptotic dimension of the free product \(*A_i\). These results are extended to include amalgamated products, \(A*_CB\), to show that \(\text{asdim}(A*_CB)<\infty\) for finite dimensional groups \(A\) and \(B\) with common subgroup \(C\).
By investigating group presentations of groups acting on graphs and trees, the authors prove that for any group \(G\) with \(\text{asdim}(G)<\infty\), an HNN extension \(G'\) of \(G\) has \(\text{asdim}(G')<\infty\). Finally, it is shown that for the case of the Davis extension, \(\Gamma\), of a group \(\pi\), if \(\text{asdim}(\pi)<\infty\), then \(\text{asdim}(\Gamma)<\infty\).

MSC:

20F69 Asymptotic properties of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20E08 Groups acting on trees
20E22 Extensions, wreath products, and other compositions of groups
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML EMIS

References:

[1] M W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. \((2)\) 117 (1983) 293 · Zbl 0531.57041 · doi:10.2307/2007079
[2] M W Davis, Coxeter groups and aspherical manifolds, Lecture Notes in Math. 1051, Springer (1984) 197 · Zbl 0543.57009
[3] M W Davis, J C Hausmann, Aspherical manifolds without smooth or PL structure, Lecture Notes in Math. 1370, Springer (1989) 135 · Zbl 0673.57020
[4] A N Dranishnikov, Asymptotic topology, Uspekhi Mat. Nauk 55 (2000) 71 · Zbl 1028.54032 · doi:10.1070/rm2000v055n06ABEH000334
[5] A Dranishnikov, On large scale properties of manifolds, · Zbl 1310.55005
[6] A N Dranishnikov, On hypereuclidean manifolds, Geom. Dedicata 117 (2006) 215 · Zbl 1095.53032 · doi:10.1007/s10711-005-9025-0
[7] A Dranishnikov, T Januszkiewicz, Every Coxeter group acts amenably on a compact space (1999) 135 · Zbl 0973.20029
[8] M Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 · Zbl 0841.20039
[9] M Gromov, Spaces and questions, Geom. Funct. Anal. (2000) 118 · Zbl 1006.53035
[10] G Mess, Examples of Poincaré duality groups, Proc. Amer. Math. Soc. 110 (1990) 1145 · Zbl 0709.57025 · doi:10.2307/2047770
[11] J Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993) · Zbl 0780.58043
[12] J P Serre, Trees, Springer (1980) · Zbl 0548.20018
[13] G Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. \((2)\) 147 (1998) 325 · Zbl 0911.19001 · doi:10.2307/121011
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.