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Introduction

  • Nick Dungey
  • A. F. M. ter Elst
  • Derek W. Robinson
Part of the Progress in Mathematics book series (PM, volume 214)

Abstract

Lie groups are manifolds symmetric under the group action and the symmetry places uniform constraints on the global properties of the manifold. The simplest constraint resulting from the group action is on the volume growth. There are only two possibilities. In the first case the volume of a ball grows no faster than a power of its radius. Groups with this characteristic are called Lie groups of polynomial growth. Compact Lie groups fall within this class since the volume is uniformly bounded. Nilpotent Lie groups also have polynomial growth, although this is less evident, and the rate of growth is straightforwardly determined by the nilpotent structure if the group is connected and simply connected. Moreover, all Lie groups of polynomial growth are unimodular. In the second case the volume of a ball grows exponentially with its radius. All non-unimodular Lie groups have exponential growth but non-unimodularity is not essential. For example, each non-compact semisimple Lie group is unimodular but has exponential volume growth.

Keywords

Heat Equation Polynomial Growth Euclidean Group Asymptotic Evolution Subelliptic Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2003

Authors and Affiliations

  • Nick Dungey
    • 1
  • A. F. M. ter Elst
    • 1
    • 2
  • Derek W. Robinson
    • 1
  1. 1.Centre for Mathematics and its Application, Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.Department of Mathematics and Computer ScienceUniversity of TechunologyEindhovenThe Netherlands

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