Peter Abramenko, Kenneth S. Brown
We study two transitivity properties for group actions on buildings, called Weyl transitivity and strong transitivity. Following hints by Tits, we give examples involving anisotropic algebraic groups to show that strong transitivity is strictly stronger than Weyl transitivity. A surprising feature of the examples is that strong transitivity holds more often than expected.
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The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all $3$-transitive actions of these groups. More precisely, suppose that $G$ is a subgroup of the homeomorphism group of the circle $\mathsf{Homeo}(\mathbb{S}^1)$ or the automorphism group of a tree $\mathsf{Aut}(\mathbb{T})$. …
Let D be a division algebra with center F and degree d>2. Let K|F be any splitting field. We analyze the action of D^\times and SL_1(D) on the spherical and affine buildings that may be associated to GL_d(K) and SL_d(K), and in particular show it is never strongly transitive. In the affine case we find examples where the action is nonetheless Weyl transitive. This extends results of Abramenko and Brown concerning the d=2 case, where strong transitivity is in fact possible. Our approach produces some explicit constructions, and we find that for d>2 the failure of the action to be strongly trans…
On the space of subgroups of Baumslag-Solitar groups II: High transitivity
本文研究Baumslag-Solitar群子群空间的高传递性，发现有限表型下高传递性仅在表型为1时普遍，无限表型下高传递性普遍除非|m|=|n|。
We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely $1$ and $\infty$. Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation represe…
An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points. This result provides a tool to rule out the existence of highly transitive actions, and to c…
For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\geq \operatorname{rank}(G)$, describes the action of the group $\operatorname{Aut}F_n$ of automorphisms of the free group $F_n$ on generating $n$-tuples of G by elementary Nielsen moves. The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property $(T)$ for $\operatorname{Aut}F_n$, $n\geq 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\ge \max\{2,\operatorname{rank}(G)\}$ when $G$ is indicable, and for $n$ big enough when $G$ is elementary amenab…
We show that the only transitive and generically $4$-transitive action of a group of finite Morley rank on a set of Morley rank $2$ is the natural action of $\operatorname{PGL}_3$ on the projective plane.
Multiple transitivity except for a system of imprimitivity
本文分类了有限块忠实的2-按块传递作用，并证明当k≥3时不存在具有非平凡块的有限块忠实k-按块传递作用。
Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups
研究在δ-双曲空间上具有一般类型作用且诱导作用忠实于Gromov边界极限集的群，根据拓扑自由性划分亚类，证明拓扑自由等价于无混合恒等式，并应用于Kac-Moody群。
We establish a sharp sufficient condition for groups acting on trees to be highly transitive when the action on the tree is minimal of general type. This gives new examples of highly transitive groups, including icc non-solvable Baumslag-Solitar groups, thus answering a question of Hull and Osin.
We study some close relationships between the classes of transitive, fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some oldstanding problems, posed by Krylov-Mikhalev-Tuganbaev in their monograph [17]. Specifically, we answer Problem 44 from there in the affirmative by constructing a Krylov transitive torsion-free Abelian group which is neither fully transitive nor transitive. This extends to the torsion-free case certain similar results in the p-torsion case, obtained by Braun et al. in J. Algebra (201…
Exceptional groups and the $s$-arc-transitivity of vertex-primitive digraphs, I
本文研究例外群对s-弧传递有向图的原始作用，证明当L为特定例外李型群时，s的上限为2。
Automorphisms of algebraic varieties and infinite transitivity
本文综述了仿射代数簇自同构群的多重与无限传递性，探讨了其与柔性的等价关系及代数几何推论，并分析了由有限单参数子群生成的情形及复分析与组合群论中的扩展应用。
We give a partial solution a question of Grigorchuk, Nekrashevych, Sushchanskii and Šunik by giving an algorithm to test whether a finite state element of an infinite iterated (permutational) wreath product $\hat G = \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr >...$ of cyclic groups of order $n$ acts spherically transitively. We can also decide whether two finite state spherically transitive elements of $\hat G$ are conjugate. For general infinite iterated wreath products, an algorithm is presented to determine whether two finite state automorphisms have the same im…
Exceptional groups and the s-arc-transitivity of vertex-primitive digraphs, II
研究基座为\(E_7(q)\)或\(E_8(q)\)的几乎单群在\(s\)-弧传递有向图上的本原作用，结合前人工作，回答了关于有限连通\(G\)-顶点本原\(s\)-弧传递有向图（非有向圈）中\(s\)是否有上界的问题。
We introduce a notion of quasimorphism between two arbitrary groups, generalizing the classical notion of Ulam. We then define and study the category of homogeneous quasigroups, whose objects are groups and whose morphisms are equivalence classes of quasimorphisms in our sense. We call the automorphism group QOut(G) of a group G in this category the quasioutomorphism group. It acts on the space of real-valued homogeneous quasimorphisms on G extending the natural action of Out(G). We discuss various classes of examples of quasioutomorphisms of free groups. We use these examples to show that the…
We prove that if the group generated by a Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth for a restricted family of automaton semigroups.
A hovel is a generalization of the Bruhat-Tits building that is associated to an almost split Kac-Moody group G over a non-Archimedean local field. In particular, G acts strongly transitively on its corresponding hovel $Δ$ as well as on the building at infinity of $Δ$, which is the twin building associated to G. In this paper we study strongly transitive actions of groups that act on affine ordered hovels $Δ$ and give necessary and sufficient conditions such that the strong transitivity of the action on $Δ$ is equivalent to the strong transitivity of the action of the group on its building at …
Schreier graphs: transitivity and coverings
本文基于群、子群和生成系刻画了Schreier图间的同构，将其视为双曲流形Mostow刚性定理的图模拟，进而给出Schreier图的传递性判据，并证明Tarski怪群满足强单性判据。
Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case
本文研究扩张 Weyl 群与加权射影线上厚子范畴的组合对应，证明管状类型下 Hurwitz 作用在 Coxeter 元约化反射分解上的传递性，从而建立保序双射。
On Minimal generating sets of splitting field, Cluster towers and Multiple transitivity of Galois groups
本文研究了分裂域中最小生成集的性质，揭示了其在Galois群多重传递性下的行为，并将其与根聚类塔概念联系起来。
Let $Ω=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $Ω$ is the integer partition $λ=(λ_1,...,λ_k)$ defined by $λ_i = |P_i|$. Let G be a group of permutations acting on $Ω$. For a fixed partition $λ$ of n, we say that G is {\em $λ$-transitive} if G has only one orbit when acting on partitions P of shape $\la$. A corresponding definition can also be given when G is just a set. For example, if $λ=(n-t,1,...,1)$, then a $λ$-transitive group is the same as a t-transitive permutation group and if $λ=(n-t,t)$, then we recover the t-homogeneous perm…
Transitivity in wreath products with symmetric groups
研究wreath积G Wr S_n中有限阿贝尔群G的传递子集结构，利用特征理论和关联方案分析，推广Livingstone-Wagner定理，并与正交多项式联系，扩展对称群S_n的研究。
Bounding the degree of generic sharp transitivity
本研究在L-群条件下证明了有限Morley秩泛型锐t-传递置换群的次数满足t≤r+2，推进了Borovik-Cherlin猜想，证明基于三个独立的群作用相关关键要素。
Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture
本文研究可定义群上的幂等Keisler测度，提出一般传递性并证明其足以发展局部稳定群论，在阿贝尔群及rosy群等新情形确立一般稳定幂等型的一般传递性，分类幂等测度，并构造可数NIP群测度卷积半群的极小左理想，证明修正的Newelski猜想。
Two generalisations of sharp k-transitivity
本文考虑群作用的尖锐\(k -\)传递性的两种推广。一是推广到尖锐\(\Theta -\)传递作用，刻画了相关条件及作用群性质，给出\(k = 2, 3\)时非分裂有限表示群例子并强化Tits结果；二是推广到关系结构的尖锐\(k -\)齐次作用，回答了Cameron的问题。
It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme …
We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan's theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing SFTs. We show that any such set of automorphisms must generate an infinite group, and als…
Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building $Δ$ and K be the stabilizer of a special vertex in $Δ$. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on $Δ$; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on $Δ$, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculia…
We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of locally 2-transitive actions. Assuming local 2-transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents. The method can be used to study Gelfand pairs and enables us to answer a question of Grigorchuk. To provide examples, we analyse in detail the local 2-transitivity of GGS-groups. Moreover, our res…
Cyclic Shuffle Groups: Universal Two-Transitivity and Complete Classification
该论文研究由k堆完美洗牌与循环堆置换生成的群H_{k,n}，证明其在n非k幂时为2-传递，完成H_{k,n}及含C_k的堆群P对应的Sh(P,n)的完全分类，解决相关群论猜想。
Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $Δ$ of dimension $\geq 2$ that acts strongly transitively on $Δ$ is Moufang. If moreover $Δ$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\G(k)^+$ of the $k$-rational points $\G(k)$ of the isotropic simple algebraic group $\G$ over a non-Archimedean local field $k$ associated with $Δ$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $Δ$, and obtain tha…
We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.
Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite permutation groups that takes as input a given system of imprimitivity for its isotropy subgroup. This produces vast families kaleidoscopic groups. We investigate their algebraic properties, such as simplicity and oligomorphy; their homological properties, such as acyclicity or contrariwise large Schur multipliers; their topological properties, such as unique…
The hyperbolic cover of an elliptic Weyl group
本文研究了椭圆韦伊系统双曲覆盖的结构及其与扩展柯西特系统的同构性，并建立了加权射影线的厚子范畴与偏序集之间的双射关系。
This paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory (2009). In this paper we determine all algebraic transformation groups $G$, defined over an algebraically closed field $\sf k$, which operate transitively, but not primitively, on a variety $Ω$, provided the following conditions are fulfilled. We ask that the (non-effective) action of $G$ on the variety of blocks is sharply 2-transitive, as well as the action on a block $Δ$ of the normalizer $G_Δ$. Also we require sharp transitivity on pairs $(X,Y)$ of i…
On a huge family of non-schurian Schur rings
本文推广H·维兰特的例子，构造了偶秩初等阿贝尔群上的一大类非舒尔型舒尔环，解决了舒尔环是否为置换群传递模的问题。
Actions of $\operatorname{Alt}(n)$ on groups of finite Morley rank without involutions
本文研究了有限 Morley 秩群上 $\operatorname{Alt}(n)$ 的忠实作用，证明了无对合时不可解群秩的下界 $n$，并推广到可解群，进而应用于通用传递性限制。
In this paper we introduce a Cayley-type graph for group-subgroup pairs and present some elementary properties of such graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these properties to those of the underlying group-subgroup pair. From the properties of the group, subgroup and generating set some of the eigenvalues can be determined, including the largest eigenvalue of the graph. In particular, when this construction results in a bipartite regular graph we show a sufficient condition on the size of the generating sets that results on R…
Linear dimension of group actions
本文研究了群作用的线性维度，通过置换模理论分析了最小维度表示的结构，并探讨了其在密码学中的应用。
We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group $\mathbb{B}_n$ on the set of $n$-adic integers $\mathbb{Z}_n$ for all natural numbers $n\geq 2$. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid $B$ an infinite sequence of braids, whose braid types are invariants of $B$. We present computations for the cases of $n=2$ and $n=3$ and use these to show th…
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