[Paper Review] Arithmeticity of the Couwenberg-Heckman-Looijenga lattices

Martin Deraux|arXiv (Cornell University)|Oct 12, 2017

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper investigates the arithmeticity of Couwenberg-Heckman-Looijenga lattices in $PU(n,1)$, proving the existence of a non-arithmetic lattice in $PU(3,1)$ that is not commensurable with the known Deligne-Mostow non-arithmetic lattice. It further computes the orbifold Euler characteristic and provides explicit presentations for all 3-dimensional examples of these lattices.

ABSTRACT

We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in $PU(n,1)$, and show that they contain a non-arithmetic lattice in $PU(3,1)$ which is not commensurable to the non-arithmetic Deligne-Mostow lattice in $PU(3,1)$. We also compute the orbifold Euler characteristic and give explicit presentations for all their 3-dimensional examples.

Motivation & Objective

• To determine the arithmetic or non-arithmetic nature of Couwenberg-Heckman-Looijenga lattices in $PU(n,1)$.
• To investigate whether any such lattice in $PU(3,1)$ is non-arithmetic and not commensurable with the Deligne-Mostow lattice.
• To compute the orbifold Euler characteristic for all 3-dimensional examples of these lattices.
• To provide explicit group presentations for all 3-dimensional Couwenberg-Heckman-Looijenga lattices.

Proposed method

• Analysis of monodromy representations and geometric structures associated with the lattices in $PU(n,1)$.
• Use of group-theoretic techniques to determine commensurability classes and arithmeticity properties.
• Computation of the orbifold Euler characteristic via topological and algebraic invariants of the associated locally symmetric spaces.
• Explicit construction of group presentations using combinatorial and geometric data from the lattice construction.

Experimental results

Research questions

• RQ1 Does the Couwenberg-Heckman-Looijenga lattice in $PU(3,1)$ admit a non-arithmetic example that is not commensurable with the Deligne-Mostow lattice?
• RQ2 What is the orbifold Euler characteristic of the 3-dimensional examples of these lattices?
• RQ3 Can explicit group presentations be derived for all 3-dimensional Couwenberg-Heckman-Looijenga lattices?
• RQ4 How do the arithmetic properties of these lattices vary across different dimensions?

Key findings

• A non-arithmetic lattice in $PU(3,1)$ is constructed that is not commensurable with the Deligne-Mostow lattice.
• The orbifold Euler characteristic is computed for all 3-dimensional examples of the Couwenberg-Heckman-Looijenga lattices.
• Explicit presentations are provided for all 3-dimensional Couwenberg-Heckman-Looijenga lattices.
• The lattice in $PU(3,1)$ is shown to be non-arithmetic and distinct in commensurability class from the Deligne-Mostow lattice.