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Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds
Edoardo Dotti1, Simon T Drewitz1, Ruth Kellerhals1
1Department of Mathematics, University of Fribourg, 1700 Fribourg, Switzerland.
Discrete & Computational Geometry
|March 8, 2023
Summary
We proved that certain elements within specific hyperbolic Coxeter 3-orbifold families are incommensurable. This incommensurability was established using cusp density, a novel analytic invariant, and its monotonic behavior.
Area of Science:
- Hyperbolic Geometry
- Group Theory
- Topology
Background:
- Non-arithmetic hyperbolic Coxeter 3-orbifolds are complex mathematical structures.
- Understanding the relationships between elements within these structures is crucial for classification.
Purpose of the Study:
- To determine the incommensurability of element pairs within three distinct infinite families of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds.
- To introduce and utilize a new analytic invariant for proving incommensurability.
Main Methods:
- Utilized the Vinberg space and Vinberg form to derive partial results.
- Introduced and analyzed the cusp density, a novel commensurability invariant.
- Leveraged the strict monotonicity of cusp density for complete proofs.
Main Results:
- Established incommensurability for pairs of elements within the same sequence in the studied orbifold families.
- Proved incommensurability for most pairs of elements belonging to different sequences.
- Demonstrated the effectiveness of cusp density as a commensurability invariant.
Conclusions:
- The study successfully proves incommensurability for specific element pairs in hyperbolic Coxeter 3-orbifolds.
- Cusp density is a powerful tool for analyzing commensurability in these geometric structures.
- The findings contribute to the understanding of non-arithmetic hyperbolic geometry.
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