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Self-representations of the Moebius group

Nicolas Monod, Pierre Py

Abstract

Contrary to the finite-dimensional case, the Moebius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations. The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy to the classical concepts of kernels of positive and negative type.

Self-representations of the Moebius group

Abstract

Contrary to the finite-dimensional case, the Moebius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations. The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy to the classical concepts of kernels of positive and negative type.

Paper Structure

This paper contains 23 sections, 21 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Formal statements
  4. Further considerations
  5. Notation and preliminaries
  6. Minkowski spaces
  7. Second model
  8. Subspaces
  9. Horospheres
  10. Kernels of hyperbolic type
  11. The notion of kernel of hyperbolic type
  12. The geometric characterization of kernels of hyperbolic type
  13. Functoriality and kernels
  14. Powers of kernels of hyperbolic type
  15. On representations arising from kernels
  16. ...and 8 more sections

Key Result

Theorem 1

For every $0<t\leq 1$ there exists a continuous self- representation $\mathrm{M\ddot ob}(E)\to \mathrm{M\ddot ob}(E)$ with the following properties:

Theorems & Definitions (43)

  • Theorem 1
  • Theorem \ref{thm:Mobnew}bis
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Definition 3.1
  • Proposition 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Example 3.6
  • ...and 33 more