Assouad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics

Higes, J.

doi:10.1090/S0002-9939-10-10240-8

Assouad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics
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by J. Higes

Proc. Amer. Math. Soc. 138 (2010), 2235-2244

DOI: https://doi.org/10.1090/S0002-9939-10-10240-8

Published electronically: February 12, 2010

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Abstract:

Suppose $G$ is a countable, not necessarily finitely generated, group. We show $G$ admits a proper, left invariant metric $d_G$ such that the Assouad-Nagata dimension of $(G,d_G)$ is infinite, provided the center of $G$ is not locally finite. As a corollary we solve two problems of A. Dranishnikov.

References

Patrice Assouad, Sur la distance de Nagata, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 1, 31–34 (French, with English summary). MR 651069
G. Bell and A. Dranishnikov, On asymptotic dimension of groups acting on trees, Geom. Dedicata 103 (2004), 89–101. MR 2034954, DOI 10.1023/B:GEOM.0000013843.53884.77
G. C. Bell, A. N. Dranishnikov, and J. E. Keesling, On a formula for the asymptotic dimension of free products, Fund. Math. 183 (2004), no. 1, 39–45. MR 2098148, DOI 10.4064/fm183-1-2
G. C. Bell and A. N. Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, Trans. Amer. Math. Soc. 358 (2006), no. 11, 4749–4764. MR 2231870, DOI 10.1090/S0002-9947-06-04088-8
G. Bell, A. Dranishnikov, Asymptotic dimension, Topology and its Appl. 155 (2008), no. 12, 1265–1296.
N. Brodskiy, J. Dydak, J. Higes, and A. Mitra, Dimension zero at all scales, Topology Appl. 154 (2007), no. 14, 2729–2740. MR 2340955, DOI 10.1016/j.topol.2007.05.006
N. Brodskiy, J. Dydak, J. Higes, A. Mitra, Nagata-Assouad dimension via Lipschitz extension, Israel J. Math. 171 (2009), 405–423.
N. Brodskiy, J. Dydak, M. Levin, A. Mitra, Hurewicz theorem for Assouad-Nagata dimension, J. Lond. Math. Soc. (2) 177 (2008), no. 3, 741–756.
N. Brodskiy, J. Dydak, U. Lang, Assouad-Nagata dimension of wreath products of groups, preprint, math.MG/0611331v2.
A. Dranishnikov, Open problems in asymptotic dimension theory, preprint, available at http://aimbri12.securesites.net/pggt.
A. Dranishnikov and J. Smith, Asymptotic dimension of discrete groups, Fund. Math. 189 (2006), no. 1, 27–34. MR 2213160, DOI 10.4064/fm189-1-2
A. N. Dranishnikov and J. Smith, On asymptotic Assouad-Nagata dimension, Topology Appl. 154 (2007), no. 4, 934–952. MR 2294641, DOI 10.1016/j.topol.2006.10.010
A. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), no. 2-3, 203–225. MR 2074917, DOI 10.1016/j.topol.2003.07.009
Cornelia Druţu, Quasi-isometry invariants and asymptotic cones, Internat. J. Algebra Comput. 12 (2002), no. 1-2, 99–135. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). MR 1902363, DOI 10.1142/S0218196702000948
J. Dydak and J. Higes, Asymptotic cones and Assouad-Nagata dimension, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2225–2233. MR 2383529, DOI 10.1090/S0002-9939-08-09149-1
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
J. Higes, Assouad-Nagata dimension of locally finite groups and asymptotic cones, preprint, math.MG0711.1512.
Urs Lang and Thilo Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 58 (2005), 3625–3655. MR 2200122, DOI 10.1155/IMRN.2005.3625
Piotr W. Nowak, On exactness and isoperimetric profiles of discrete groups, J. Funct. Anal. 243 (2007), no. 1, 323–344. MR 2291440, DOI 10.1016/j.jfa.2006.10.011
D. Osin, Asymptotic dimension of relatively hyperbolic groups, Int. Math. Res. Not. 35 (2005), 2143–2161. MR 2181790, DOI 10.1155/IMRN.2005.2143
John Roe, Hyperbolic groups have finite asymptotic dimension, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2489–2490. MR 2146189, DOI 10.1090/S0002-9939-05-08138-4
Yehuda Shalom, Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math. 192 (2004), no. 2, 119–185. MR 2096453, DOI 10.1007/BF02392739
J. Smith, On asymptotic dimension of countable abelian groups, Topology Appl. 153 (2006), no. 12, 2047–2054. MR 2237596, DOI 10.1016/j.topol.2005.07.011
Justin Smith, The asymptotic dimension of the first Grigorchuk group is infinity, Rev. Mat. Complut. 20 (2007), no. 1, 119–121. MR 2310581, DOI 10.5209/rev_{R}EMA.2007.v20.n1.16546
Kevin Whyte, Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture, Duke Math. J. 99 (1999), no. 1, 93–112. MR 1700742, DOI 10.1215/S0012-7094-99-09904-0