## Notes of John Parker’s Cambridge lecture 25-04-2017

Non arithmetic lattices

With Martin Deraux and Julien Paupert.

1. Arithmetic lattices

We are interested in lattices in ${G=PU(2,1)}$. One source of examples is the following arithmetic construction due to Borel-Harish Chandra. Let ${\mathbb{G}}$ be an algebraic linear group defined over ${{\mathbb Q}}$. Assume there exists a surjective homomorphism ${\phi:\mathbb{G}_{\mathbb R}\rightarrow G}$ with compact kernel. Then any subgroup ${\Gamma commensurable with ${\phi(\mathbb{G}_{\mathbb Z})}$ is called arithmetic. It is a lattice. We are interested in other lattices.

In ${SO(n,1)}$, there are infinitely many commensurability classes of non-arithmetic lattices (even reflection groups if ${n=2}$). In ${SU(n,1)}$, ${n\geq 4}$, no example is known. In ${SU(3,1)}$, there is 1 example. In ${SU(2,1)}$, a handful of examples were obtained by Mostow in 1980 (one of which was rediscovered by Ron Livne), and these remained the only known examples until the work I will describe now.

2. Complex reflection groups

In ${H^2_{\mathbb C}}$, an isometry fixing a complex line rotates in the perpendicular direction. If the angle is ${2\pi/p}$, ${p\in{\mathbb N}}$, ${p\geq 2}$, we speak of a complex reflection.

We investigate groups generated by three complex reflections with equal, fixed, angles. Such triples depends on 4 real parameters: one may think of them as the three angles and the Toledo invariant of a triangle.

Let ${A}$, ${B}$ be complex reflections. Studying discreteness of the subgroup they generate leads to the braid relation: there exists an integer ${q}$ (called braiding length) such that

$\displaystyle \begin{array}{rcl} (AB)^{q/2}=(BA)^{q/2} \end{array}$

(when ${q}$ is odd, this means one keeps only one letter: for instance, if ${n=3}$, ${ABA=BAB}$). This restricts the “angle” parameters to numbers of the form ${2\cos(\pi/q)}$, ${q=a,b,c}$. We also introduce the braiding length ${n}$ of the pair ${(R_1, R_3^{-1}R_2 R_3)}$.

We expect that if the group ${\langle R_1,R_2,R_3 \rangle}$ is a lattice, then the following elements are non-loxodromic,

$\displaystyle \begin{array}{rcl} 12,23,31,123\bar 2,1\bar 3 23, 12\bar 1 3,123. \end{array}$

We are unable to prove that this is necessary (nor sufficient), but empirical experience supports it.

2.1. Arithmeticity criterion

Let ${L}$ be a purely imaginary quadratic extension of a totally real number field ${k}$. Let ${H}$ be a Hermitian form defined over ${L}$, of signature ${(n,1)}$.

Proposition 1 (Mostow, after Takeuchi and Vinberg) Let ${\Gamma}$ be a lattice in ${SU(H,\mathcal{O}_L)}$. Then ${\Gamma}$ is arithmetic iff for all Galois automorphisms ${\sigma}$ of ${L}$, not inducing identity on the field of traces ${{\mathbb Q}[\mathrm{trace}(Ad(\Gamma)]}$, ${^\sigma H}$ is definite.

2.2. Building fundamental domains

The basic building block is a pyramid, i.e. a triangle and its images by the subgroup fixing one vertex. ${123}$ has a single fixed point, we want it inside our fundamental domain. ${123}$ maps the pyramid to another one that fits along the base.

Next we need fill in faces. Instead of using projective geometry (as Rick Schwartz did in his RHOCHI paper), we use bissectors in complex hyperbolic geometry (however, our fundamental domains are not Dirichlet). To check that this produces a polyhedron, some verifications are done by exact computations over number fields, performed by the computer.

2.3. Output

We recover Deligne-Mostow’s examples and find a few more: the number of commensurability classes passes from 13 to 22.

The ground field ${k}$ has low degree (2, 3,…).

We have not tried quadrilaterals yet…