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Axial distances in discrete Möbius groups.

F W Gehring1, G J Martin

  • 1Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.

Proceedings of the National Academy of Sciences of the United States of America
|March 15, 1992
PubMed
Summary

New lower bounds for the trace of a commutator in discrete groups of Möbius transformations were established. These findings provide sharp estimates for distances between elliptic axes, impacting hyperbolic manifold and orbifold volume calculations.

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Area of Science:

  • * Mathematics
  • * Geometric Group Theory
  • * Hyperbolic Geometry

Background:

  • * Discrete groups of Möbius transformations are fundamental in geometry and topology.
  • * Understanding the geometry of these groups, particularly the distance between axes of elliptic elements, is crucial.
  • * Previous bounds for the trace of the commutator and related geometric quantities were limited.

Purpose of the Study:

  • * To establish new lower bounds for the trace of the commutator of a discrete two-generator group of Möbius transformations.
  • * To derive sharp estimates for the distance between the axes of elliptic elements within such groups.
  • * To lay the groundwork for new bounds on the volume of hyperbolic manifolds and orbifolds.

Main Methods:

  • * Analysis of the algebraic structure of discrete two-generator groups of Möbius transformations.

Related Experiment Videos

  • * Application of techniques to derive lower bounds for the trace of the commutator.
  • * Geometric interpretation of algebraic results to estimate distances between axes of elliptic elements.

    • Main Results:

    • * Novel lower bounds for the trace of the commutator of discrete two-generator Möbius transformation groups.
    • * Sharp estimates for the distance between the axes of elliptic elements in these groups.
    • * Foundation for forthcoming results on hyperbolic manifold and orbifold volumes.

    Conclusions:

    • * The derived bounds offer significant improvements in estimating geometric quantities related to discrete Möbius groups.
    • * These results have direct implications for understanding the volume of hyperbolic manifolds and orbifolds.
    • * The study advances the geometric analysis of discrete groups and their associated spaces.