Cone types and geodesic languages for lamplighter groups and Thompson's group F

群的类点灯者几何

Lamplighter-like geometry of groups

本文提出 halo 积以推广圈积并统一多类点灯者型群，同时构建几何框架，利用精细不变量刻画其大尺度几何并区分拟等距类型。

Seesaw words in Thompson's group F

We describe a family of words in Thompson's group F which present a challenge to the question of finding canonical minimal length representatives, and which show that F is not combable by geodesics. These words have the property that there are only two possible suffixes of long lengths for geodesic paths to the word from the identity; one is of the form $g^k$ and the other of the form $g^{-k}$ where g is a generator of the group.

Infinitely supported Liouville measures of Schreier graphs

We provide equivalent conditions for Liouville property of actions of groups. As an application, we show that there is a Liouville measure for the action of the Thompson group $F$ on dyadic rationals. This result should be compared with a recent result of Kaimanovich, where he shows that the action of the Thompson group F on dyadic rationals is not Liouville for all finitely supported measures. As another application we show that there is a Liouville measure for lamplighter actions. This gives more examples of non-amenable Liouville actions.

关于灯夫群的有限扩张

On finite extensions of lamplighter groups

本文通过构造灯夫群的有限扩张，系统回答了组合群论中关于子群成员资格、字问题、共轭问题等多个开放问题，并研究了余字问题、剩余有限性和同构问题。

Geodesics and Visual boundary of Horospherical Products

We study the geometry of horospherical products by providing a description of their distances, geodesics and visual boundary. These products contains both discrete and continuous examples, including Cayley graphs of lamplighter groups and solvable Lie groups of the form IR semi-direct product with (N1 xN2), where N1 and N2 are two simply connected, nilpotent Lie groups.

Counting elements and geodesics in Thompson's group $F$

We present two quite different algorithms to compute the number of elements in the sphere of radius $n$ of Thompson's group $F$ with standard generating set. The first of these requires exponential time and polynomial space, but additionally computes the number of geodesics and is generalisable to many other groups. The second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius $n$ with $n \leq 1500$. Using the resulting series data we find that the growth rate of the group is bounded above by $2.62167...$. This is very close to Guba's lower …

On the cogrowth of Thompson's group $F$

We investigate the cogrowth and distribution of geodesics in R. Thompson's group $F$.

点灯人小组具有无限弱警察数

The Lamplighter groups have infinite weak cop number

针对局部有限顶点传递图弱警察数是否存在非1和∞值的开放问题，研究 wreath 积 Cayley 图，证明非平凡群与无限群 wreath 积及 Thompson 群$F$的 Cayley 图具无限弱警察数，解决部分问题。

超几何作用于汤普森群F及其推广

Hyperbolic actions of Thompson's group $F$ and generalizations

研究超几何结构在汤普森群F及其推广F_n上的偏序结构，揭示其局部复杂性与全局简单性，证明超几何结构的丰富子偏序及与lamplike结构的关联。

Restrictions on Potential Automatic Structures on Thompson's Group F

We show that a large class of languages in the standard finite generating set X = {$x_0, x_1, x_0^{-1}, x_1^{-1}$} cannot be part of an automatic structure for Thompson's Group F. These languages are ones that accept at least one representative of each element of F of word length that is within a fixed constant of a geodesic representative of the element. To accomplish this, we look at a specific element of F and trace two different paths through the Cayley graph to that element. We show that staying within the length restrictions along these two paths would force that element to have contradi…

Thompson's group $F$ is not Liouville

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 ${\mathbb R}$). …

Metric Properties of Diestel-Leader Groups

In this paper we investigate metric properties of the groups $Γ_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $Γ_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}…

Thompson组及相关组的数值研究

Numerical studies of Thompson's group F and related groups

本文研究了Thompson组及相关组的共增长级数，开发了多项式时间算法生成这些级数的项，并通过数值技术提取其渐进行为，给出了Thompson组F的共增长级数的改进下界，探讨了该组是否可 amenable 的问题。

$\mathcal C$-graph automatic groups

We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov (arXiv:1107.3645) by replacing the regular languages in their definition with more powerful language classes. For a fixed language class $\mathcal C$, we call the resulting groups $\mathcal C$-graph automatic. We prove that the class of $\mathcal C$-graph automatic groups is closed under change of generating set, direct and free product for certain classes $\mathcal C$. We show that for quasi-realtime counter-graph automatic groups where normal forms have length that is linear in the geod…