We prove a strong form of model completeness for expansions of the field of real numbers by (the real and imaginary parts of) the modular function J, by the modular forms E4 and E6 and quasimodular form E2 defined in the usual fundamental domain, and the restricted sine function and the (unrestricted) exponential function. This is done using ideas of Peterzil and Starchenko’s paper [12] on the ...

Every surface in the Euclidean space R3 admits a canonical Riemannian metric that has constant Gaussian curvature and is conformal to the original metric. Similarly, 3manifolds can be decomposed into pieces that admit canonical metrics. Such metrics not only have theoretical significance in 3-manifold geometry and topology, but also have potential applications to practical problems in engineeri...

Let Γ be a cyclic subgroup of PSL2(C) generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of Γ on HR are the complements of configurations of half-balls centered on the plane at infinity ∂HR. Jørgensen (On cyclic groups of Möbius transformations, Math. Scand. 33 (1973), 250–260) proved that the boundary of the intersection of the Ford fundamental domain ...

Deligne and Mostow constructed a class of lattices in PU(2, 1) using monodromy of hypergeometric functions. Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for the lattices with three fold symmetry in the list of Deligne and Mostow. This is a generalisation of the works of Boadi and Parker and gives a different interpretation ...

In this paper some properties of quadratic forms whose base points lie in the point set F Π , the fundamental domain of the modular group, and transforming these forms into the reduced forms with the same discriminant ∆ < 0 are given.

Though parts of the present paper perfectly look like abstract general nonsense, its origin is a rather concrete question. Given a simply connected almost simple Chevalley group G and a field k, what is a fundamental domain (in an appropriate sense) for the action of G(k[t, t−1]) on the product ∆+ × ∆− of the two associated Bruhat–Tits buildings ∆+ = ∆(G(k((t−1)))) and ∆− = ∆(G(k((t)))) ? The s...

We prove that the symbolic dynamical system generated by a purely substitutive Arnoux-Rauzy sequence is measurably conjugate to a toral translation. The proof is based on an explicit construction of a fundamental domain with fractal boundary (a Rauzy fractal) for this toral translation. Communicated by Pierre Liardet Dedicated to the memory of Gérard Rauzy

This is a survey of our recent joint investigations of lattices that are generated by finite Abelian groups. In the case of cyclic groups, the volume of a fundamental domain of such a lattice is a perturbed Toeplitz determinant with a simple Fisher-Hartwig symbol. For general groups, the situation is more complicated, but it can still be tackled by pure matrix theory. Our main result on the lat...

Poincaré’s classical theorem of fundamental polygons is a widely known, valuable tool that gives sufficient conditions for a (convex) hyperbolic polygon, equipped with so-called side-pairing transformations, to be a fundamental domain for a discrete subgroup of isometries. Poincaré first published the theorem in dimension two in 1882. In the past century, there have been several published proof...

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that...