Non-arithmetic lattices and the Klein quartic

The Klein Quartic

Figure 1: Fundamental region R of the Modular Group, given by {z ∈ H : |z| > 1,− 2 ≤ Re(z) ≤ 1 2} The Klein Quartic curve is famous for its symmetries and many visualizations are created to capture these in lower dimensions. In fact, it has the largest possible automorphism group for its genus, the simple group of order 168. Yet, interest in this algebraic curve started with knowledge of the ex...

The Klein Quartic in Number Theory

We describe the Klein quartic X and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of X; an explicit identification of X with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat.

New non-arithmetic complex hyperbolic lattices

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 construc...

Polyhedral models of Felix Klein ́s Quartic

Quasi-isometric rigidity of non-cocompact S-arithmetic lattices

Throughout we let K be an algebraic number field, VK the set of all inequivalent valuations on K, and V ∞ K ⊆ VK the subset of archimedean valuations. We will use S to denote a finite subset of VK that contains V ∞ K , and we write the corresponding ring of S-integers in K as OS. In this paper, G will always be a connected non-commutative absolutely simple algebraic K-group. Any group of the fo...