Note on the Cantor-Bendixson rank of limit groups
英文摘要:
We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group.
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Abderezak Ould Houcine
We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We study the Cantor--Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for $G$ either the Grigorchuk group or the Gupta--Sidki $3$ group, the Cantor--Bendixson rank of $\mathrm{Sub}(G)$ is $ω$. For each natural number $n$, we additionally characterize the subgroups of rank $n$ and give a description of subgroups in the perfect kernel.
Space of orders with finite Cantor-Bendixson rank
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We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (G_n) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank omega^n.
We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism.
Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the space of pointed Schreier graphs of the action of $F$ on the orbits of dyadic rational numbers and corresponding Schreier dynamical system are described. In particular, we answer the question of Grigorchuk on the Cantor-Bendixson rank of the underlying space of the Schreier dynamical system in the context of $F$. As applications we prove that the pointed Schreier gra…
Perfect kernel and dynamics: from Bass-Serre theory to hyperbolic groups
本文通过引入新方法研究可数群子群空间的Cantor-Bendixson分解与完美核,确定了许多新群的完美核和秩,建立了共轭作用的拓扑传递性条件,并由此得到类A中大量新的群例,涵盖直角Artin群、极限群、小消去群、随机群及虚拟紧特殊群等。
Algebraic and geometric properties of homeomorphism groups of ordinals
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The purpose of this note is to prove that the set of connective constants of Cayley graphs contains a Cantor space.
A natural topology on the set of left orderings on free abelian groups and free groups $F_n$, $n>1$ has studied in [1]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a conjecture: this is also true for the free group $F_n$ with $n>1$ generators. We point out the article dealing with equivalent questions. The answer is "yes".
Distortion in groups of generalized piecewise-linear transformations
研究广义分段线性变换群随Cantor-Bendixson秩参数递增形成的升链中元素畸变概念的变化,构造出在低秩群无畸变、高秩群有畸变的元素,揭示畸变性质对群秩的依赖关系。
The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups if different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fu…
A normal subgroup of the (extended) mapping class group of a surface is said to be geometric if its automorphism group is the mapping class group. We prove that in the case of the Cantor tree surface, every normal subgroup is geometric. We note that there is no non-trivial finite-type mapping class group for which this statement is true. We study a generalisation of the curve graph, proving that its automorphism group is again the mapping class group. This strategy is adapted from that of Brendle-Margalit and the author for certain normal subgroups in the finite-type setting.
Homeomorphism groups of self-similar 2-manifolds
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These notes are the English version of the paper "Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a …
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