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Assouad–Nagata dimension of connected Lie groups

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Abstract

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.

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Authors and Affiliations

  1. Institute of Mathematics, MA 6-2, TU Berlin, 10623, Berlin, Germany

    J. Higes

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    I. Peng

Authors
  1. J. Higes
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  2. I. Peng
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Correspondence to J. Higes.

Additional information

J. Higes is supported by project MEC, MTM2006-0825 and ‘contrato flechado’ i-math.

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Higes, J., Peng, I. Assouad–Nagata dimension of connected Lie groups. Math. Z. 273, 283–302 (2013). https://doi.org/10.1007/s00209-012-1004-1

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  • DOI: https://doi.org/10.1007/s00209-012-1004-1

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