Aperiodic colorings and tilings of Coxeter groups

Embedding of hyperbolic Coxeter groups into products of binary trees and aperiodic tilings

We prove that a finitely generated, right-angled, hyperbolic Coxeter group can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of the group. As application we obtain certain strongly aperiodic tilings of the Davis complex of these groups.

Aperiodic tilings of manifolds of intermediate growth

We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.

Homological aperiodic tilings of 3-dimensional geometries

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds appearing in the geometrization conjecture.

Subtilings of Elnitsky Tilings for Finite Irreducible Coxeter Groups

Two new Elnitsky tilings for Coxeter groups of type $\mathrm{B}$ are displayed as certain subtilings. Additionally, a new tiling for the non-crystallographic Coxeter group of type $\mathrm{H}_3$ is obtained, described as a $\mathrm{D}_6$-subtiling.

有限Coxeter群与广义Elnitsky铺砌

Finite Coxeter Groups and Generalized Elnitsky Tilings

本文基于Elnitsky对A、B、D型Coxeter群约化词与多边形铺砌的双射，通过嵌入对称群的方法，将其推广至所有有限Coxeter群。

有限Coxeter群的Bruhat序与Elnitsky铺砌

The Bruhat Order of a Finite Coxeter Group and Elnitsky Tilings

本文证明有限Coxeter群中任意元素及其约化表达式对应一个2m-多边形的Elnitsky铺砌，其中m为子群指数，构造性证明结合E-嵌入与删除序，并给出E8等示例。