Axial distances in discrete Möbius groups

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.
• * 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.