Universal groups for point-sets and tilings
نویسندگان
چکیده
We study the universal groups of inverse semigroups associated with point sets and with tilings. We focus our attention on two classes of examples. The first class consists of point sets which are obtained by a cut and projection scheme (so-called model sets). Here we introduce another inverse semigroup which is given in terms of the defining data of the projection scheme and related to the model set by the empire congruence. The second class is given by one-dimensional tilings.
منابع مشابه
Aperiodic Tilings , Positive Scalar Curvature , and Amenability of Spaces
The object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity. Manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers. We shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties. The most interesting result characterizes when H ~f...
متن کاملCohomology groups for projection point patterns
Aperiodic point sets (or tilings) which can be obtained by the method of cut and projection from higher dimensional periodic sets play an important role for the description of quasicrystals. Their topological invariants can be computed using the higher dimensional periodic structure. We report on the results obtained for the cohomology groups of projection point patterns supplemented by explici...
متن کاملDense Dirac combs in Euclidean space with pure point diffraction
Regular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally compact Abelian groups, is the pure pointedness of the diffraction spectrum. We derive an extension of this result, valid for dense point sets in Euclidean s...
متن کاملTiling Groups with Difference Sets
We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and inve...
متن کاملCrystallography and Riemann Surfaces
The level set of an elliptic function is a doubly periodic point set in C. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in C2 and its sections (“cuts”) by C. We give S a crystallographic isometry in C2 by defining a fundamental surface element as a conformal map of triangular domains and S as its extension by reflections in the triangle edg...
متن کامل