Crystallographic Groups, Strictly Tessellating Polytopes, and Analytic Eigenfunctions

Strictly Convergent Analytic Structures

We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on p-adic subanalytic sets, and we continue the study of nonarchimedean fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the language LK consisting of the language of valued fields together with all strictly convergent p...

Zeros of real analytic ergodic Laplace eigenfunctions

Orbifold Triangulations and Crystallographic Groups

Given a triangulation of a 3-dimensional euclidean orbifold, e.g. in terms of the Delaney symbol of a periodic tiling, a method is discussed for identifying the isomorphism type of the corresponding space group. Of the 219 types of groups, 175 can be recognized solely by considering the orbifold graph associated with the given triangulation. Simple algebraic invariants distinguish between the r...

Fibered Orbifolds and Crystallographic Groups

In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group Γ determines a geometric fibered orbifold structure on the flat orbifold En/Γ, and conversely every geometric fibered orbifold structure on En/Γ is determined by a normal subgroup N of Γ, which is maximal in its commensurability class of normal subgroups of Γ. In particular, we prove that En/Γ is a fiber...

Cocompactly cubulated crystallographic groups

We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually Z group is isomorphic to the hyperoctahedral triangulation of Sn−1, providing a class of groups G for which the simplicial boundary of a G-cocompact cube complex depends only on G. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension n...