Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups

幂零 Cantor 作用的素谱与动力学

Prime spectrum and dynamics for nilpotent Cantor actions

本文研究了幂零 Cantor 作用的稳定性，证明了具有有限素谱的作用必稳定，并构造了不可数个具有无限素谱且不稳定的 Heisenberg 群作用。

Existence and genericity of finite topological generating sets for homeomorphism groups

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite topological generating sets in related groups. We show that the generic pair of elements in the homeomorphism group $Homeo_+(I)$ generate a dense subgroup of $Homeo_+(I)$. By contrast, if $M$ is any compact connected manifold with boundary other than the interval, we observe that a…

Dynamics of induced homeomorphisms of one-dimensional solenoids

We study the displacement function of homeomorphisms isotopic to the identity of the universal one-dimensional solenoid and we get a characterization of the lifting property for an open and dense subgroup of the isotopy component of the identity. The dynamics of an element in this subgroup is also described using rotation theory.

Smoothing nilpotent actions on 1-manifolds

Let $M$ be a connected 1-manifold, i.e., $M = \R \cong (0, 1), [0, 1), [0, 1]$, or $S^1$, and let $\Homeo_+(M)$ (resp. $\Diff_+^1(M)$) be the group of orientation-preserving homeomorphisms (resp. $C^1$ diffeomorphisms) of $M$. It is a classical result that if $N$ is a finitely-generated, torsion-free nilpotent group, then there exist 1-1 homomorphisms $ϕ\colon N \to \Homeo_+(M)$. Farb and Franks show that, in fact, there exists a 1-1 homomorphism $N \to \Diff_+^1(M)$. In this paper we obtain a stronger result: every action $ϕ\colon N \to \Homeo_+(M)$ is topologically conjugate to an action $\t…

Finite orbits for nilpotent actions on the torus

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.

Normal subgroups of diffeomorphism and homeomorphism groups of R^n and other open manifolds

We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

Estimation of recurrence for nilpotent group action

We estimate size of recurrence of an action of a nilpotent group by homeomorphisms of a compact space for polynomial mappings into a nilpotent group form the partial semigroup $(\mathcal{P}_{f}(\mathbb{N}),\uplus)$. To do this we have used algebraic structure of the Stone-Čech copactification partial semigroup and that of the given nilpotent group.

nilpotent群作用于一维紧致流形的临界正则性

Critical regularity of nilpotent groups acting on one-dimensional compact manifolds

研究nilpotent群在闭区间或半开区间及圆上的忠实微分作用的临界正则性，通过代数方法推导出表达式，推广了先前成果。

同胚群作用的分类复杂性

Classification complexity of homeomorphism group actions

本文探讨紧致可度量化空间的同胚群被稠密非闭子群取代时轨道等价关系分类复杂性的变化，分析三类子群及两类空间的超空间作用，发现其复杂性与闭子群不同。

大轨道的直向极限与Knaster连续统的同胚群

Direct limits of large orbits and the Knaster continuum homeomorphism group

本研究证明泛Knaster连续统的一度同胚开子群具有贫集补共轭类，通过Polish群作用中贫集补轨道的直向极限稠密逼近一般性结论，结合区间保向同胚群共轭作用的精细分析得到主定理。

Groups of diffeomorphisms of one-manifolds, III: Nilpotent subgroups

Plante-Thurston proved that every nilpotent subgroup of $\Diff^2(S^1)$ is abelian. One of our main results is a sharp converse: $\Diff^1(S^1)$ contains every finitely-generated, torsion-free nilpotent group.

Chain groups of homeomorphisms of the interval

We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson's group $F$. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. On the other hand, every finitely generated subgroup of $\operatorname{Homeo}^+(I)$ can be realized as a subgr…

Fixed points of nilpotent actions on ${\mathbb S}^{2}$

We prove that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks, Handel and Parwani for the abelian case. We show that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point. M…

Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups

For a group hyperbolic relative to virtually nilpotent subgroups, on a cusped graph associated to the group, we construct a random walk whose Martin boundary is the Bowditch boundary of the group. Moreover, the harmonic measure is a conformal density corresponding to a hyperbolic Green metric and is exact dimensional on the Bowditch boundary. The latter equipped with a visual distance induced by the Green metric is an Ahlfors-regular metric measure space. The dimension is given in terms of the drift, a Green drift and the asymptotic entropy. The Patterson-Sullivan density for the action on the…

嵌入直线的康托尔集同胚的随机作用与Tits二择性

Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative

该研究类比Margulis关于圆周同胚群的Tits二择性结论，证明了实数集紧子集的局部单调同胚群也满足类似的二择性质，拓展了同胚群领域的相关理论结果。

关于区间上具有有限多个不动点的同胚群

On groups of homeomorphisms of the interval with finitely many fixed points

本文证明含两生成元自由半群的Diff₊(I)子群非C₀-离散，并借此推广Hölder定理，完成对区间同胚群中每个非单位元至多有N个不动点的子群的分类。

On the simplicity of homeomorphism groups of a tilable lamination

We show that the identity component of the group of homeomorphisms that preserve all leaves of a R^d-tilable lamination is simple. Moreover, in the one dimensional case, we show that this group is uniformly perfect. We obtain a similar result for a dense subgroup of homeomorphisms.

Anosov子群的幂零覆盖上的扭曲Patterson-Sullivan密度

Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

本文为Anosov子群构造扭曲共形密度并证明其唯一性、无原子性和遍历性，进而利用商群特征分类幂零覆盖上正规子群的遍历共形密度。

The homeomorphism group of the universal Knaster continuum

We define a projective Fraissé family whose limit approximates the universal Knaster continuum. The family is such that the group $\textrm{Aut}(\mathbb{K})$ of automorphisms of the Fraissé limit is a dense subgroup of the group, $\textrm{Homeo}(K)$, of homeomorphisms of the universal Knaster continuum. We prove that both $\textrm{Aut}(\mathbb{K})$ and $\textrm{Homeo}(K)$ have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extreme…

Non-realizability of the pure braid group as area-preserving homeomorphisms

Let $\text{Homeo}_+(D^2_n)$ be the group of orientation-preserving homeomorphisms of $D^2$ fixing the boundary pointwise and $n$ marked points as a set. Nielsen realization problem for the braid group asks whether the natural projection $p_n:\text{Homeo}_+(D^2_n)\to B_n:=π_0(\text{Homeo}_+(D^2_n))$ has a section over subgroups of $B_n$. All of the previous methods either use torsions or Thurston stability, which do not apply to the pure braid group $PB_n$, the subgroup of $B_n$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside th…