有限阿贝尔群中瓦片（tile）的周期性

阿贝尔群作用的连续组合数学

Continuous Combinatorics of Abelian Group Actions

本文发展了基于超非周期元素构造的技术，用于分析阿贝尔群连续作用生成的等价关系理论，证明了 $F(2^{\mathbb{Z}^2})$ 的连续色数恰好为 4，并通过瓦片分析将连续同态的存在性问题归约为有限组合问题。

Tiling by lattices for locally compact abelian groups

For a locally compact abelian group $G$ a simple proof is given for the known fact that a bounded domain $Ω$ tiles $G$ with translations by a lattice $Λ$ if and only if the set of characters of $G$ indexed by the dual lattice of $Λ$ is an orthogonal basis of $L^2(Ω).$ The proof uses simple techniques from Harmonic Analysis.

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

For an ascending HNN-extension $G*_ψ$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_ψ}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-c…

非交换群上的双射差问题

A nonabelian twist on differences of bijections

本文研究非交换有限群中双射差问题，探讨多集合可表示为双射的商形式的条件，提出基于循环铺砌的结构结果。

Cayley Graph Expanders and Groups of Finite Width

We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only 2 generators and 4 relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring M(3,F2). We present explicit vector space bases for the finite abelian quotients of the lower exponent-2 groups of G by upper tri…