Matthew G. Brin
We show that Thompson's group F occurs with great frequency in the group of PL homeomorphisms of the unit interval.
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We show that for a large class $\mathcal{C}$ of finitely generated groups of orientation preserving homeomorphisms of the real line, the following holds: Given a group $G$ of rank $k$ in $\mathcal{C}$, there is a sequence of $k$-markings $(G, S_n), n\in \mathbf{N}$ whose limit in the space of marked groups is the free group of rank $k$ with the standard marking. The class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain "self-similarity" type hypothesis. Examples include Thompson's group $F$, Higman-Thompson groups, Stein-Thompson groups, var…
We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G)) = \mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group $\mathcal{R}_{2}$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))= \mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Th…
We look at the automorphisms of Thompson type groups of piecewise linear homeomorphisms of the real line or circle that use slopes that are integral powers of a fixed integer n with n>2. We show that large numbers of "exotic" automorphisms appear---automorphisms that are represented as conjugation by non-PL homeomorphisms of the real line or circle. This is in contrast to the n=2 case where no such automorphisms appear.
This paper allows one to obtain a criterion for the existence of a projectively invariant measure formulated in terms of combinatorial properties of a group (amenability of some canonical quotient group). Such necessary and sufficient condition is a basis of the classification scheme for groups of homeomorphisms of the line. In particular, a nonamenability of Thompson's group $F$ follows from the obtained criterion.
Irreducible fast sets of bump homeomorphisms generate copies of Thompson's groups $F_n$
研究由不可约几何快速正凸起集生成的群，证明对于\(n\geq2\)，由\(n\)个此类集合生成的群同构于\(n\)元汤普森群\(F_n\)，回答了布林和扎雷姆斯基提出问题的强版本。
We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. As a first application, we demonstrate that any coherent group action $G<Homeo^+(\mathbf{R})$ that produces a nonamenable equivalence rela…
Higman--Thompson groups $F_n$ all the way down
证明对$n\geq2$，希格曼 - 汤普森群$F_n$有无限指数且同构于自身的极大子群。构造相关子群链，刻画了特定同胚，给出将共轭问题转化为算法的机制，并应用于解决一些群同构问题。
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。
The Thompson group V is a subgroup of the homeomorphism group of the Cantor set. Brin defined higher dimensional Thompson groups nV as generalizations of V. We prove that nV has Serre's property FA, for every n. This is a generalization of the corresponding result of Farley, who studied V.
Let T denote Thompson's group of piecewise 2-adic linear homeomorphisms of the circle. Ghys and Sergiescu showed that the rotation number of every element of T is rational, but their proof is very indirect. We give here a short, direct proof using train tracks, which generalizes to PL homeomorphism of the circle with rational break points and derivatives which are powers of some fixed integer, and also to certain flows on surfaces which we call "Thompson-like". We also obtain an explicit upper bound on the smallest period of a fixed point in terms of data which can be read off from the combina…
We describe, through the use of Rubin's theorem, the automorphism groups of the Higman-Thompson groups $G_{n,r}$ as groups of specific homeomorphisms of Cantor spaces $\mathfrak{C}_{n,r}$. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: to characterise the automorphism groups of the `Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of $V$, which was the `last chameleon.' The home…
We complete the program begun by Brin and Squier of characterising conjugacy in Thompson's group $F$ using the standard action of $F$ as a group of piecewise linear homeomorphisms of the unit interval.
This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's group F and suitable larger groups of piecewise-linear homeomorphisms of the unit interval. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at a particular diagram associated to an element. In the second part, we describe the structure of subgroups of the group of all homeomor…
Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.
In his papers [2], [3] Brin introduced the higher dimensional Thompson groups nV which are generalizations to the Thompson's group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n = 2. We show how to generalize his construction to obtain a finite presentation for every positive integer n. As a corollary, we obtain another proof that the groups nV are simple (first proved by Brin in [4]).
We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.
We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…
The automorphism groups of several of Thompson's countable groups of piecewise linear homeomorphisms of the line and circle are computed and it is shown that the outer automorphism groups of these groups are relatively small. These results can be interpreted as stability results for certain structures of PL functions on the circle. Machinery is developed to relate the structures on the circle to corresponding structures on the line.
For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are stabilizers of the minimal set. This fact is an important element in the classification theorem, used, in particular, in the study of the Thompson's group $F$.
We determine the abstract commensurator com(F) of Thompson's group F and describe it in terms of piecewise linear homeomorphisms of the real line and in terms of tree pair diagrams. We show com (F) is not finitely generated and determine which subgroups of finite index in F are isomorphic to F. We show that the natural map from the commensurator group to the quasi-isometry group of F is injective.
We construct a "higher dimensional" version 2V of Thompson's group V. Like V it is an infinite, finitely presented, simple subgroup of the homeomorphism group of the Cantor set, but we show that it is not isomorphic to V by showing that the actions on the Cantor set are not topologically conjugate: 2V has an element with "chaotic" action, while V cannot have such an element. A theorem of Rubin is then applied which shows that for these two groups, isomorphism would imply topological conjugacy.
A Thompson group for the basilica
本文构造了 basilica Julia 集上的 Thompson 型同胚群，引入弧对图表证明其有限生成性，并证明了该群几乎单。
Finitely presented simple left-orderable groups in the landscape of Richard Thompson's groups
本文首次构造了实数线上保向同胚的有限表示单群，该类群具有类型 $F_{\infty}$、无限几何维数及非平凡齐次拟态，从而具有无限换位子宽度。
Notes that elaborate on the details of a Theorem of Shavgulidze that implies the amenability of R. J. Thompson's group F.
$\mathrm{C}^*$-selflessness of groups of homeomorphisms
本文证明具有忠实分段极小-极近作用的可数群是$\text{C}^*$-无私的，并推广到拓扑全群，涵盖希格曼-汤普森群和布林-汤普森群。
The set of finitely generated subgroups of the group $PL_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R.~Thompson's group $F$. In this paper we show that every finitely generated subgroup $G<PL_+(I)$ is either soluble, or contains an embedded copy of Brin's group $B$, a finitely generated, non-soluble group, which verifies a conjecture of the first author from 2009. In the case that $G$ is soluble, we show that the derived length of $G$ is bounded above by the number of breakpoints of any finite set of generat…
We study the properties of rotation numbers for some groups of piecewise linear homeomorphisms of the circle. We use these properties to obtain results on PL rigidity, non isomorphicity, non exoticity of automorphisms, non smoothability for Thompson-Stein groups.
Richard Thompson's group F is the group of piecewise linear homeomorphisms of the unit interval with a finite number of break points, all at dyadic rational numbers (their denominators are powers of 2) and with slopes which are powers of 2. A discrete group G is amenable if there exists a finitely-additive probability measure on G which is invariant under left translations and is defined on all subsets of G. The amenability question for F is a well known open problem. In this paper we prove that group F is not amenable.
We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.
In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing the elements of the group by labeled tree diagrams in a manner which closely parallels Richard Thompson's group F.
Limit groups with respect to Thompson's group $F$ and other hereditarily separating groups
本文证明Thompson群$F$上的极限群不能是$F$关于有限生成子群的HNN-扩张，但给出了中心化HNN-扩张的例子，并研究了相对极限群。
The subgroup structure of Thompson's group F is not yet fully understood. The group F is a subgroup of the group PL(I) of orientation preserving, piecewise linear self homeomorphisms of the unit interval and this larger group thus also has a poorly understood subgroup structure. It is reasonable to guess that F is the "only" subgroup of PL(I) that is not elementary amenable. In this paper, we explore the complexity of the elementary amenable subgroups of F in an attempt to understand the boundary between the elementary amenable subgroups and the non-elementary amenable. We construct an example…
On the group of homeomorphisms of the Basilica
本文证明了Basilica分形的同胚群等价于某非局部有限带状树的变换群，并指出Basilica Thompson群在其保向同胚群中稠密。
Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompson's group F thus: G consists of all orientation preserving PL-homeomorphisms of the rea…
We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $\mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's "golden ratio" group $F_τ$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the …
Thompson's group F is the group of all increasing dyadic piecewise linear homeomorphisms of the closed unit interval. We compute Sigma^m(F) and Sigma^m(F;Z), the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F, and we show that Sigma^m(F) = Sigma^m(F;Z). As an application, we show that, for every m, F has subgroups of type F_{m-1} which are not of type F_{m}.
Every copy of Thompson's group $F$ in $F$ is undistorted
该研究否定回答了Thompson群$F$是否存在自身的畸变拷贝的问题，证明$F$中每个同构于$F$的子群都是无畸变的。
We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $\mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criteria for embedding subgroups of $\mathrm{Homeo}_+(I)$ into Richard Thompson's group $F$. In particular, every member of our class of generating sets generates a group which embed…
We study two--generated subgroups $\langle f,g\rangle<\mathrm{Homeo}^+(I)$ such that $\langle f^2,g^2\rangle$ is isomorphic to Thompson's group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with nonabelian free subgroups, examples which do not admit faithful actions by $C^2$ diffeomorphisms on $1$--manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to M.…
Semi-simple actions of the Higman-Thompson groups $T_n$ on finite-dimensional CAT(0) spaces
本研究将汤普森群$T$的半单作用全局不动点结论推广到其推广群Higman-Thompson群$T_n$，借助环群视角证明$T_n$在有限覆盖维数完备CAT(0)空间上的半单作用均有全局不动点。
Action graphs, semiconjugacy, and non-embedding in Thompson's group $V$
通过证明有限生成子群的作用图拟等距于树，并建立一类实线同胚群嵌入$V$时半共轭于标准作用，从而证明Stein群$F_{2,3}$等群不能嵌入$V$。
In a recent paper, we showed that groups admitting "veelike actions" on a finite language embed in mapping class groups of certain two-sided subshifts. In this note, we illustrate this theorem for the embedding of Thompson's $V$ by exhibiting the piecewise linear local rules for the embedding. These turn out to split the embedding into the mapping class group, showing that $V$ even embeds in the mapping class group by homeomorphisms.
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。
Just infinite quotients of finitely generated subgroups of $PL^+[0,1]$
本文证明 Richard Thompson 群 F 及两类分段同胚群的有限生成子群的恰无限商实际上交换，从而回答 Grigorchuk 的问题，并给出众多非顺从群例子。
Given a group $G$ of homeomorphism of a first-countable Hausdorff space $\mathcal{X}$, we prove that if the action of $G$ on $\mathcal{X}$ is minimal and has rigid stabilisers that act locally minimally, then the neighbourhood stabilisers of any two points in $\mathcal{X}$ are conjugated by a homeomorphism of $\mathcal{X}$. This allows us to study stabilisers of points in many classes of groups, such as topological full groups of Cantor minimal systems, Thompson groups, branch groups, and groups acting on trees with almost prescribed local actions.
In this partly expository paper, we study the set A of groups of orientation-preserving homeomorphisms of the circle S^1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in A. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T .
On a family of finitely generated simple groups of homeomorphisms of the circle
本文研究环群换位子群的单性，构造出具有Aℝ性质的环群，证实圆周保定向同胚群中存在不可数个有限生成简单群。
Let $PL_0(I)$ represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group $W$ and show that $W$ embeds in every non-solvable subgroup of $PL_0(I)$. We find mild conditions under which other non-solvable subgroups ($B$, $(\wr\Z\wr)^{\infty}$, $(\Z\wr)^{\infty}$, and $^{\infty}(\wr\Z)$) embed in subgroups of $PL_0(I)$. We show that all solvable subgroups of $PL_0(I)$ embed in all non-solvable subgroups of $PL_0(I)$. These results continue to apply if …
Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.
Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated
研究Cantor空间上同胚群的vigorous和flawless性质，证明简单vigorous群要么由 torsion元生成，要么非可数生成；vigorous群简单当且仅当flawless，且vigorous简单子群类广泛。
We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group PLo(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and also of the solvable subgroups of the generalized R. Thompson groups F_n. The first description is as a set of isomorphism classes of groups which is closed under three algebraic operations, and the second is as the set of isomorphism classes of subgroups of a countable collection of easily described groups. We show the two descriptions are equivalent.
We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of the generalized Thompson groups F_n. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the F_n. We also show that any standard restricted wreath product C wr T (of non-trivial groups) that embeds in PLo(I) or F_n must have T isomorphi…
We investigate subgroups of the group PLo(I) of piecewise-linear, orientation preserving homeomorphisms of the unit interval with finitely many breaks in slope, and also subgroups of Thompson's group F. We find geometric criteria determining the derived length of any such group, and use this criteria to classify the solvable and non-solvable subgroups of PLo(I) and of F. Let H be a subgroup of PL_o(I) or F. We find that H is solvable if and only if H is isomorphic to a group in a well described class R of groups. We also find that H is non-solvable if and only if we can embed a copy of a speci…
Topological Full Groups of Irreducible Edge Shifts have Solvable Conjugacy Problem
本研究利用链图和交换环半群等方法，解决了不可约边移位的拓扑全群的共轭问题，相关技术可推广至包含Houghton群、类Thompson群$QV$的更大边移位分段典范同胚群类。
Conjugacy problems in higher-dimensional Thompson groups
该研究证明了n≥2时Brin-Thompson群$nV$中元素的拓扑共轭性不可判定，还得到了Cantor空间同胚群有限生成超群中Whitehead问题及群论共轭的不可判定性。
The Poisson boundary of Thompson's group $T$ is not the circle
本文证明了在特定条件下，赋予平稳测度的圆周$S^1$不是$(G,μ)$的Poisson边界，从而回答了关于Thompson群$T$的Poisson边界是否为圆周的开放问题。
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 ${\mathbb R}$). …
The Zariski Topology on Homeomorphism groups
本文研究了同胚群上的扎里斯基拓扑，证明了 Thompson 群 F 和 T 的扎里斯基拓扑与标准紧开拓扑一致，而 V 群的扎里斯基拓扑不可约，不满足 Hausdorff 或群拓扑条件。
In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in H_1}π(g)=\sum_{h\in H_2}π(h)$ and such that the closed ideal generated by $\sum_{g\in H_1}λ(g)-\sum_{h\in H_2}λ(h)$ coincides with $C^*_λ(T)$, then the Richard Thompson group $F$ is not amenable. In particular, if $C_λ^*(T)$ is simple then $F$ is not amenable. Here we prove the converse, namely, if $F$ is not amenable then we can find two sets $H_1$ and $H_2$ with the above pro…
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。
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