Let G be a connected linear Lie group which finitely covers the isometry group of X. Furthermore, let Γ ⊂ G be a discrete subgroup. We assume that Γ is geometrically finite. We refer to Definition 2.1 for a precise explanation of this notion. If X is a real hyperbolic space, then Γ is geometrically finite iff it admits a fundamental domain with finitely many totally geodesic faces. In the other...

We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight k for SL2(Z) interlace on most of the lower boundary of the fundamental domain.

This paper is concerned with a priori C∞ regularity for threedimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimens...

Let Sf be a compact Riemann surface, which we will assume to have curvature normalized to be — 1 , and let 0=A0<A1^A2^• • be the eigenvalues corresponding to the problem AF+AF=0 on Sf \ where A is the Laplacian for £P. In an otherwise very interesting and useful paper [2], McKean has stated that it is always the case that A ^ J. In this paper, we will show that this need not be true, and that i...

We give a general structure theory for reconstructing group actions on sets without any assumptions on the group, the action, or the set on which the group acts. Using certain ‘local data’ D from the action we build a group G(D) of the data and a space X (D) with an action of G(D) on X (D) that arise naturally from the data D. We use these to obtain an approximation to the original group G, the...

Let Λ0, . . . , Λn ⊂ R be a finite collection of lattices of the same volume. When does there exist a set Ω ⊂ R that is a fundamental domain for all Λi, i = 0, . . . , n? The main result of this paper is that when the (group-theoretic) sum Λ0 + · · · + Λn of the dual lattices is direct (this means that the equation x0 + · · · + xn = 0, with xi ∈ Λi , has no nontrivial solution), then a Borel me...

Resolving the incompleteness of Weil-Petersson metric on Teichmüller spaces by taking metric and geodesic completion results in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller space as its non...

Let K be a locally compact hypergroup. In this paper we initiate the concept of fundamental domain in locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fun...

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L for any p∈ [1; +∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody latti...