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The Margulis Invariant of Isometric Actions on Minkowski (2+1)-Space

In: Rigidity in Dynamics and Geometry

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  • William M. Goldman

    (University of Maryland, Department of Mathematics)

Abstract

Let denote an affine space modelled on Minkowski (2+1)-space and let Γ be a group of isometries whose linear part (Γ) is a purely hyperbolic subgroup of SO0(2,1). Margulis has defined an invariant α: Γ → ℝ closely related to dynamical properties of the action of Γ. This paper surveys various properties of this invariant. It is interpreted in terms of deformations of hyperbolic structures on surfaces. Proper affine actions determine deformations of hyperbolic surfaces in which all the closed geodesics lengthen (or shorten). Formulas are derived showing that α grows linearly on along a coset of a hyperbolic one-parameter subgroup. An example of a deformation of hyperbolic surfaces is given along with the corresponding Margulis space-time.

Suggested Citation

  • William M. Goldman, 2002. "The Margulis Invariant of Isometric Actions on Minkowski (2+1)-Space," Springer Books, in: Marc Burger & Alessandra Iozzi (ed.), Rigidity in Dynamics and Geometry, pages 187-201, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04743-9_9
    DOI: 10.1007/978-3-662-04743-9_9
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