Hyperbolic Coxeter groups with Sierpiński carpet boundary

Constructing Bowditch boundaries of some relatively hyperbolic groups that are homeomorphic to the $n$-dimensional Sierpiński carpet

In this paper we prove that if some relatively hyperbolic groups have Bowditch boundary homeomorphic to the $n$-sphere, then they are also relatively hyperbolic with respect to another set of parabolic subgroups and its Bowditch boundary is homeomorphic to the $n-1$-dimensional Sierpiński carpet.

Quasisymmetric rigidity of Sierpinski carpets $F_{n,p}$

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group. We also establish that $F_{n,p}$ and $F_{n',p'}$ are quasisymmetrically equivalent if and only if $(n,p)=(n',p')$.

不相交型 Sierpinski 地毯双曲分支是有界的

Sierpinski carpet hyperbolic components of disjoint type are bounded

本文通过建立不相交型双曲分支的一致先验界，证明不相交型 Sierpinski 地毯双曲分支有界，并给出其闭包上映射的拟二次限制及 $z^2$ 边界后临界集描述。

Relations between topological and metrical properties of self-affine Sierpi$\acute{\text{n}}$ski sponges

We investigate two Lipschitz invariants of metric spaces defined by $δ$-connected components, called the maximal power law property and the perfectly disconnectedness. The first property has been studied in literature for some self-similar sets and Bedford-McMullen carpets, while the second property seems to be new. For a self-affine Sierpi$\acute{\text{n}}$ski sponge $E$, we first show that $E$ satisfies the maximal power law if and only if $E$ and all its major projections contain trivial connected components; secondly, we show that $E$ is perfectly disconnected if and only if $E$ and all it…

具有一维边界的双曲群

Hyperbolic groups with 1-dimensional boundary

若挠自由双曲群G有1维边界，则只要G不在循环群上分裂，该边界是Menger曲线或Sierpinski地毯.当G的边界是Sierpinski地毯时，证明了G是三维双曲Poincare对偶群的拟凸子群。我们还构造了一个“拓扑刚性”双曲群G：G的边界的任何同胚都是由G的一个元素诱导的。

Nonhyperbolic Coxeter groups with Menger curve boundary (with erratum)

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on $n$ vertices for $n\geq 5$. The construction depends on a slight extension of Sierpiński's theorem on embedding $1$--dimensional planar compacta into the Sierpiński carpet. See the appendix for a brief erratum.