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Simple Tilings of Nilpotent Lie Groups

Kyle Hansen

Abstract

We define simple tilings in the general context of a G-tiling on a Riemannian homogeneous space M to be tilings by "almost linear" simplices. As evidence that this definition is natural, we prove that a natural class of tilings of M are MLD to simple ones. We demonstrate the utility of this definition by generalizing previously known results about simple tilings of Euclidean space. In particular, it is shown that a simple tiling space of a connected, simply connected, rational, nilpotent Lie group is homeomorphic to a rational tiling space, and therefore a fiber bundle over a nilmanifold. We further sketch a proof of the fact that there is an isomorphism between Čech cohomology and pattern equivariant cohomology of simple tilings in connected, simply connected, nilpotent Lie groups.

Simple Tilings of Nilpotent Lie Groups

Abstract

We define simple tilings in the general context of a G-tiling on a Riemannian homogeneous space M to be tilings by "almost linear" simplices. As evidence that this definition is natural, we prove that a natural class of tilings of M are MLD to simple ones. We demonstrate the utility of this definition by generalizing previously known results about simple tilings of Euclidean space. In particular, it is shown that a simple tiling space of a connected, simply connected, rational, nilpotent Lie group is homeomorphic to a rational tiling space, and therefore a fiber bundle over a nilmanifold. We further sketch a proof of the fact that there is an isomorphism between Čech cohomology and pattern equivariant cohomology of simple tilings in connected, simply connected, nilpotent Lie groups.
Paper Structure (20 sections, 29 theorems, 39 equations, 1 figure)

This paper contains 20 sections, 29 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Tiling Spaces
  4. Basic Preliminaries
  5. Gähler Complexes and Inverse Limits
  6. Notions of Equivalence
  7. Simple Tilings in Euclidean Space
  8. Examples
  9. Delaunay Sets and Triangulations
  10. Delaunay Sets and Complexes
  11. Delaunay Triangulations of Manifolds
  12. Simple Tilings
  13. Nilpotent Lie Groups
  14. Notation and Foundations
  15. The Heisenberg Group
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a Riemannian homogeneous space. Given a geometrically normal $G$-tiling $\mathcal{T}$ of $\mathcal{M}$, there is a simple tiling $\mathcal{T}_{\Delta}$ which is MLD to $\mathcal{T}$.

Figures (1)

  • Figure 1: When perturbing the Delaunay set in the $\theta_1$-neighborhood of the $1$-cell $XZ$, the points in the $r$-neighborhood of the yellow zones are generic by induction, and so these points will not change when we apply the extended algorithm to this neighborhood. In particular, since the $\theta_1$-neighborhoods of $XY$ and $XZ$ lie inside the ball $\eta$ of radius $\frac{\theta_0}{2}$, their Delaunay sets inside this intersection agree. These neighborhoods are far enough away outside of $\eta$ so that their union contains no degeneracy after applying the extended algorithm.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 2.1: sadun2003tilinginverselimits
  • Theorem 2.2: priebe2000towards
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 3.1: boissonnat2018delaunay
  • Theorem 3.2: boissonnat2013stability
  • ...and 43 more