Emina Alibegovic
We prove that all elements of infinite order in $Out(F_n)$ have positive translation lengths; moreover, they are bounded away from zero. Consequences include a new proof that solvable subgroups of $Out(F_n)$ are finitely generated and virtually abelian and the new result that such subgroups are quasi-convex.
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Translation length formula for two-generated groups acting on trees
本文提出了两生成群作用于树上的翻译长度公式，并通过伪长度公理推导出相关结论，为自由群的自同构轨道研究提供新方法。
Fibrations of 3-manifolds and asymptotic translation length in the arc complex
本文研究了3-流形纤维化变化时弧复形中伪Anosov单值性渐近平移长度的变化，定义了规范化函数$\mu_d$，发现其图像聚点集仅取决于切片形状且性质良好，并指出给出该函数公式极为困难。
Given a quasi-median graph $X$, the crossing graph $ΔX$ and the contact graph $ΓX$ are natural hyperbolic models of $X$. In this article, we show that the asymptotic translation length in $ΔX$ or $ΓX$ of an isometry of $X$ is always rational. Moreover, if $X$ is hyperbolic, these rational numbers can be written with denominators bounded above uniformly; this is not true in full generality. Finally, we show that, if the quasi-median graph $X$ is constructible in some sense, then there exists an algorithm computing the translation length of every computable isometry. Our results encompass contac…
Random walks and contracting elements II: Translation length and Quasi-isometric embedding
本文研究具有压缩元素的度量空间上的随机游走，证明等距群的随机子群拟等距嵌入，并建立压缩元素的普遍性及平移长度的中心极限定理及其逆定理。
Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\textrm{Out}(F_n)$, of the free group of rank $n$ is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group $\textrm{Out}(G)$ of a finitely generated free-by-finite group $G$. We show that $\textrm{Out}(G)$ is translation discrete and that every subgroup of $\textrm{Out}(G)$ is either virtually finitely generated abelian or contains a nonabelian free group.
Entropy and Translation Length in the Ray Graph
本文证明平面去掉康托集的映射类群在射线图上的平移长度受拓扑熵除以log(2)控制，通过组合与几何论证结合，并猜想光滑性条件可去除。
We define a family of four-point invariants for Shilov boundaries of bounded symmetric domains of tube type, which generalizes the classical four-point cross ratio on the unit circle. This generalization, which is based on a similar construction of Clerc and Ørsted, is functorial and well-behaved under products; these two properties determine our extension uniquely. Our generalized cross ratios can be used to estimate translation lengths of a large class of isometries of the underlying bounded symmetric domain. Our main application concerns maximal representations of surface groups with Hermit…
Terminal Geometry of Rank-Two Nielsen Reduction on Simplicial Trees
本文通过连续分数和轴几何方法，研究了树上Schottky对的生成条件，揭示了超几何自同构体生成自由离散子群的几何不变量及其与三-gap定理的关系。
Let F_n be a free group of rank n>1. Two elements g, h in F_n are said to be translation equivalent in F_n if the cyclic length of ϕ(g) equals the cyclic length of ϕ(h) for every automorphism ϕof F_n. Let F(a, b) be the free group generated by {a, b} and let w(a,b) be an arbitrary word in F(a,b). We prove that w(g, h) and w(h, g) are translation equivalent in F_n whenever g, h \in F_n are translation equivalent in F_n, which hereby gives an affirmative solution to problem F38b in the online version (http://www.grouptheory.info) of [1].
Let F_2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for given two elements u, v of F_2, u and v are translation equivalent in F_2, that is, whether or not u and v have the property that the cyclic length of phi(u) equals the cyclic length of phi(v) for every automorphism phi of F_2. This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [1] for the case of F_2.
Relative Free Splitting Complexes III: Stable Translation Lengths and Filling Paths
研究相对自由分裂复形及其外自同构群作用，通过研究填充路径，得到稳定平移长度与相对轨道跟踪动力学的定量关系，包括特定轨道条件下有填充吸引层叠及相应的正下界，结果对$\text{Out}(F_n)$也是新的。
A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of B. Farb.
Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem
研究相对自由分裂复形\(\mathcal{FS}(\Gamma;\mathscr A)\)中由相对外自同构诱导的单纯等距同构的稳定平移长度，通过‘两个所有定理’得出沿折叠路径的指数扩张性质，应用该定理得到从相对外空间到分裂复形的映射性质及稳定平移长度上界。
Let $γ_0$ be a curve on a surface $Σ$ of genus $g$ and with $r$ boundary components and let $π_1(Σ)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $γ$ of type $γ_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $π_1(Σ)$ the limit $$\lim_{L\to\infty}\frac 1{L^{6g-6+2r}}\{γ\text{ of type }γ_0\text{ with }S\text{-translation length}\le L\}$$ exists and is positive. The main new technical tool is that the function which associates t…
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