New non-arithmetic complex hyperbolic lattices (1401.0308v4)

Abstract: We produce a family of new, non arithmetic lattices in PU(2,1). All previously known examples were commensurable with lattices constructed by Picard, Mostow and Deligne-Mostow, and fell into 9 commensurability classes. Our groups produce 5 new distinct commensurability classes. Most of the techniques are completely general, and provide efficient geometric and computational tools for constructing fundamental domains for discrete group acting on the complex hyperbolic plane.

• The paper introduces five novel commensurability classes of non-arithmetic complex hyperbolic lattices in PU(2,1) through explicit fundamental domain constructions.
• It employs geometric and computational methods, including the Poincaré polyhedron theorem, to rigorously establish the lattice properties.
• Quantitative measures such as Euler characteristics and volume calculations highlight the distinct non-arithmetic features compared to classical Deligne–Mostow examples.

New Non-Arithmetic Complex Hyperbolic Lattices: An Analysis

The paper by Martin Deraux, John R. Parker, and Julien Paupert explores the construction of new, non-arithmetic complex hyperbolic lattices within the group ${\rm PU}(2,1)$. Historically, the exploration into non-arithmetic complex hyperbolic lattices has been dominated by examples falling into nine commensurability classes, largely influenced by the work of Picard, Mostow, and Deligne–Mostow. The authors of this paper advance the field by introducing five distinct commensurability classes through their construction of novel non-arithmetic lattices.

Novel Construction Techniques

The authors utilize geometric and computational methods to systematically construct fundamental domains for discrete groups operating in complex hyperbolic spaces. Their approach leverages complex reflection groups, extending Mostow's previous work. A noteworthy aspect of their methodology is the explicit construction of fundamental polyhedra using the Poincaré polyhedron theorem, which serves as a critical tool in demonstrating the lattice properties of the groups they paper.

Implications for Classification of Lattices

The established classification of arithmetic groups is rooted in number-theoretical data, but non-arithmetic lattices present a more complex scenario. The presented lattices, adding new commensurability classes, underscore the diversity possible when engaging with ${\rm PU}(2,1)$, expanding the known landscape beyond the previously established nine classes.

Mathematical Framework and Results

• Complex Reflection Groups: The authors focus on groups generated by complex reflections and regular elliptic elements, demonstrating that they are lattices by rigorously constructing fundamental domains.
• Commensurability Classes: The research brings forth five new commensurability classes in ${\rm PU}(2,1)$, which are not equivalent to any previously known Deligne–Mostow lattices.
• Euler Characteristics and Volume Calculations: Through detailed combinatorial analysis, the authors calculate the orbifold Euler characteristics, which range from $2/63$ for $p=3$ to $221/1008$ for $p=12$, offering quantitative insights into the geometry and topology of the corresponding quotients.

Non-arithmetic vs. Arithmetic Properties

One of the pivotal results is that among these newly constructed lattices, only the one with $p=3$ is arithmetic. The apparent non-arithmetic nature of the other lattices is underscored by examining the adjoint trace fields, which differ from those of arithmetic Deligne–Mostow lattices. The authors effectively utilize Galois conjugation properties and trace field computations to highlight the non-arithmetic characteristics.

Future Directions in Complex Hyperbolic Geometry

The paper speculates potential future avenues in the paper of non-arithmetic lattices, driven by their construction techniques. These developments could facilitate exploration in higher dimensions or more complex structures, potentially revealing further non-arithmetic forms not yet conceptualized.

In summary, this paper potentially shifts the understanding of complex hyperbolic lattices, presenting a set of tools and insights that can aid in further exploration of non-arithmetic lattices, reinforcing complex hyperbolic geometry's critical role in modern mathematical research.