Gili Golan Polak
We prove that under two natural probabilistic models (studied by Cleary, Elder, Rechnitzer and Taback), the probability that a random pair of elements of Thompson's group $F$ generate the entire group is positive. We also prove that for any $k$-generated subgroup $H$ of $F$ which contains a "natural" copy of $F$, the probability of a random $(k+2)$-generated subgroup of $F$ coinciding with $H$ is positive.
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A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated.
On the generation problem in Thompson's groups $F_n$
通过子群的核与闭包以及相关自动机，研究Higman-Thompson群$F_n$的生成问题，给出子集生成$F_n$的充分条件及有限情形下的验证算法，并构造了$F_n$中一个无不动点的无限指数极大子群，同构于$F_{2n-1}$。
We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogue way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic …
Computational explorations in Thompson's group F
我们描述了在Thompson群F中的一些计算探索的结果。我们描述的实验，以估计其标准的有限生成集，旨在解决的微妙和困难的问题是否汤普森的组是顺从的共同增长的F。我们还描述了实验来估计的指数增长性率的F和逃逸率的对称随机游动相对于标准生成集。
The goal is to extend a non-standard amenability test for groups, based on random walks and superharmonic functions, to group actions on sets, and to apply it to Thompson's group F using certain properties of extensive amenability. While no conclusive answer regarding the amenability of F is given, the approach is helpful in developing a new potentially useful criterion and testing it on a significant subclass of superharmonic functions.
Boundary of the action of Thompson group F on dyadic numbers
证明了$F$作用于$\mathbb{D}$的Schreier图上的简单随机游动的Poisson边界是非平凡的，其中$\mathbb{D}$是$[0, 1]$中的并元数的集合.这给出了Kaimanovich的结果：Thompson群$F$不具有Liouville性质的新证明.此外，我们还计算了$F$作用于$\mathbb{D}$的Schreier图的增长函数.
Random subgroups of Thompson's group $F$
我们考虑Thompson群$F$的随机子群相对于所有$k$生成子群的集合的两个自然分层。我们发现，对于这两种分层，出现正密度的子群的同构类是不一样的。我们给出了第一个已知的例子{ \em持久}子群，其同构类发生的正密度内的一组$k$生成子群，所有足够大的$k$。此外，汤普森群提供了第一个没有子群的类属同构类的群的例子。$F$的元素由有限根二叉树的约简对唯一地表示。我们计算的渐近增长率和生成函数的数量减少对树木，我们表明是D-有限的，而不是代数。然后，我们使用渐近增长来证明我们的密度结果。
On Thompson Groups for Ważewski Dendrites
本研究针对作用于Ważewski枝形曲线的类汤普森重排群展开，分析了其有限生成性、在全同胚群中的稠密性等性质，还推广得到Airplane重排群$T_A$，发现这类群与汤普森群$V$特征更相似。
Length-based cryptanalysis: The case of Thompson's Group
基于长度的方法是一种启发式的解决随机生成的方程组，具有合理的表现长度函数。我们描述了几个改进以前建议的基于长度的算法，使他们适用于汤普森的组显着的成功率。特别地，这表明基于Thompson群的Shpilrain-Ushakov公钥密码系统是不安全的，并且表明基于此群的实用公钥密码系统不可能是安全的。
Invariable generation of certain groups of piecewise projective homeomorphisms of the real line
本文证明了实数直线上分段射影同胚群、分段PSL(2,Z)同胚群、Monod群H(Z)、分段PSL(2,Q)同胚群（有理断点）以及Higman-Thompson群F_n（n≥3）和黄金比例Thompson群F_τ均为恒定生成群。
We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). …
We introduce a new distributional invariance principle, called `partial spreadability', which emerges from the representation theory of the Thompson monoid $F^+$ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid $F^+$. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a …
The one-point Schreier Poisson boundary of Thompson's group $F$
通过将Thompson群$F$上的简单对称随机游走投影到二进轨道点$1/2$，识别单点Schreier链随机游走的Poisson边界为骨架端边界，并证明击中测度是带显式偏置的Bernoulli乘积测度。
Generation and Simplicity in the Airplane Rearrangement Group
研究了Airplane极限空间的重排群$T_A$，证明了其由Thompson群$F$和$T$生成且有限生成，并揭示了其换位子群的单性、2-传递性及与Basilica重排群的包含关系。
The "spread" of Thompson's group $F$
针对 Thompson 群 $F$ 不能 $\frac{3}{2}$-生成但近乎满足该性质的背景，论文研究其 spread 与 uniform spread 概念的修正版本。
Finite generation for the group $F\left(\frac32\right)$
研究证明 Thompson 型群 $F\left(\frac32\right)$ 可由两个元素有限生成，明确了其断点在 $\mathbb{Z}[\frac16]$、斜率为 $\frac32$ 的幂时的生成结构。
Generating simple vigorous groups
研究简单活跃群的生成问题，通过构造性证明得出该类群的多个强生成结果，如由三个对合生成等，证明汤普森群V是(2,3)生成的，还强化了相关嵌入定理，建立了新的生成准则。
We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we prove that either $H$ is solvable, or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson's group $F$ that generate it as a semigroup and have…
Houghton-like groups from "shift-similar" groups
本文定义移位相似群及由其构造的类 Houghton 群 H_n(G)，证明有限生成、顺从性等结果，并表明任意有限生成群均可嵌入有限生成移位相似群。
We consider a transitive action of a finitely generated group $G$ and the Schreier graph $Γ$ defined by this action for some fixed generating set. For a probability measure $μ$ on $G$ with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of $Γ$. As a corollary we obtain that for a probability measure with a finite first moment on Thompson's group $F$, the support of which generates $F$ as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is…
We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.
Concentration inequalities for maximal displacement of random walks on groups of polynomial growth
本文研究了具有多项式增长的群上随机游走最大位移的集中不等式，证明了在特定群类上随机游走的扩散性质。
Random sampling of trivials words in finitely presented groups
我们描述了一种新的算法，随机抽样的自由减少的话等于身份在一个共同提出的组。该算法基于Metropolis Monte Carlo抽样。该算法从拉伸玻尔兹曼分布\begin{align*}π(w) &= (|w|+1)^α β^{|w|} \cdot Z^{-1} \end{align*}采样，其中$|w|$是字的长度$w$，$α$和$β$是算法的参数，并且$Z$是归一化常数。因此，相同长度的字以相同的概率被采样。该分布可以表示在一系列的共生组，这使我们能够相关的统计特性的算法采样的单词的共生组，因此其顺从性。我们已经实现了该算法，并将其应用到几个组介绍，包括Baumslag-Solitar组，一些自由积Kouksov研究，一个不服从次指数的顺从组（基于巴西利卡组），和理查德·汤普森的组$F$。
Piecewise isometry groups of Euclidean tessellations
该研究证明了欧几里得空间中与晶体根系相关的足够好的镶嵌的分段等距群是初等可和的，其后续工作将证明这类群的有限生成性和更高有限性性质。
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