manuscripta mathematica

, Volume 54, Issue 1, pp 17-52

Old and new on S1(2)

  • Joachim HilgertAffiliated withFachbereich Mathematik, Technische Hochschule Darmstadt
  • , Karl H. HofmannAffiliated withFachbereich Mathematik, Technische Hochschule Darmstadt

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The structure of the three dimensional Lie group (2,ℝ) and its universal covering group G is surveyed in an explicit fashion with detailed computational and geometrical information on their exponential functions and one parameter groups. In particular, a new global parametrisation of the group G is given which allows a convenient description of the exponential function and its singularities.

This information is applied to give a rather complete theory of infinitesimally generated subsemigroups both in S1(2) and in G. In this context we exihibit the exceptional role played by the semigroup Sl(2)+ of all Sl(2)-matrices with non-negative entries and the semigroup Σ+ in G which is generated by one of the two cones in (2,ℝ) containing the elements which give non-positive values to the CartanKilling form.