Mustafa Gokhan Benli, Rostislav Grigorchuk, Tatiana Nagnibeda
We exhibit examples of groups of intermediate growth with $2^{\aleph_0}$ ergodic, continuous, invariant random subgroups. The examples are the universal groups associated with a family of groups of intermediate growth.
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This article describes a practical approach for determining the lattice of subgroups U < V < G between given subgroups U and G, provided the total number of such subgroups is not too large. It builds on existing functionality for element conjugacy, double cosets and maximal subgroups.
In this paper we characterize the finite permutation groups $F<S_d$ on $d$ letters such that every compact open subgroup of the associated universal group $U(F)<{\rm Aut} T_d$ is topologically finitely generated. Actually we show that in this case the groups are positively finitely generated.
P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.
We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a group and relative quasiconvexity for subgroups of the group.
We show that every Grigorchuk group $G_ω$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; it can also have finitely generated infinite torsion subgroups, as well as residually finite subgroups that are not elementary amenable, answering questions of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_ω$ can be embedded in a finitely generated simple group that…
Given a class C of subgroups of a topological group G, we say that a subgroup H in C is a universal C subgroup of G if every subgroup K in C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural pre-order on the set of subgroups of G. We show that for any locally compact Polish group G, the countable power of G has a universal K-sigma subgroup and a universal compactly generated subgroup. We prove a weaker version of this in the non-locally compact case and provide an example showing that this result cannot readily be impro…
Maximal subgroups of groups of intermediate growth
本文解决了作用于有根二叉树的中间增长群的无穷指数最大子群数量问题，证明帐篷映射的非挠迭代单项群恰有可数无穷多个此类子群，并证实Šunić定义的所有群均具有同余子群性质且恰好无穷。
We prove that in a simple real Lie group, there is no Borel measurable dense subgroup of intermediate Hausdorff dimension.
For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept to consider quotients of inverse semigroup. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quo…
Some properties of abnormal subgroups in generalized soluble groups will be considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it will be proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H coincides with its normalizer in G. This result will extend on radical groups the well-known criterion of abnormality for finite soluble groups obtained by D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugations will be also proven.
We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and …
We construct a moduli space for the connected subgroups of an algebraic group and the corresponding universal family. Morphisms from an algebraic variety to this moduli space correspond to flat families of connected algebraic subgroups parametrised by this variety. This moduli space is obtained by gluing together infinitely many irreducible projective varieties of bounded dimension along closed subvarieties. Regarding families of non-connected subgroups of an algebraic group, we show that, given sich a family, the corresponding family of identity components is an irreducible component of the f…
We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.
For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G) \leq 1/6$ for every nonsolvable group $G$.
We prove that every noncyclic subgroup of a free $m$-generator Burnside group $B(m,n)$ of odd exponent $n \gg 1$ contains a subgroup $H$ isomorphic to a free Burnside group $B(\infty,n)$ of exponent $n$ and countably infinite rank such that for every normal subgroup $K$ of $H$ the normal closure $<K >^{B(m,n)}$ of $K$ in $B(m,n)$ meets $H$ in $K$. This implies that every noncyclic subgroup of $B(m,n)$ is SQ-universal in the class of groups of exponent $n$.
This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of $S_{\infty}$. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings, and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows to provide a partial answer to a question posed by Kechris, Pestov and Todorcevic.
It is shown that FC-central extensions retain sub-exponential volume growth. A large collection of FC-central extensions of the first Grigorchuk group is provided by the constructions in the works of Erschler and Kassabov-Pak. We show that in these examples subgroup separability is preserved. We introduce two new collections of extensions of the Grigorchuk group. One collection gives first examples of intermediate growth groups with centers isomorphic to $\mathbb{Z}^{\infty}$; and the other provides groups with prescribed oscillating intermediate growth functions.
Uni-width subgroups, universal elements, and lambda number of finite groups
本文研究了有限群的单宽子群、通用元素和lambda数，证明了幂图存在非恒等通用元素的条件，并给出了lambda数的最优界。
Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ denote a non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalized quaternion. We determine the universal deformation rings $R(S_5,V)$ and $R(\hat{S}_5,V)$ for each mod 2 representation $V$ of $S_5$ that belongs to the principal 2-modular block of $S_5$ and whose stable endomorphism ring is given by scalars when it is inflated to $\hat{S}_5$. We show that for these $V$, a question raised by the first author and Chinburg concerning the relation of the universal deformation ring of $V$ to the Sylow 2-subgroups…
We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all natural number k a torsion group with growth function asymptotically $\exp(n^{1-(1-α)^k}),\quad 2^{3-3/α}+2^{2-2/α}+2^{1-1/α}=2$, and a torsion-free group with growth function asymptotically $\exp(\log(n)n^{1-(1-α)^k})$. These are the first examples of groups of intermediate growth for which the growth function is known. We construct a group of intermediate growth that contains the group of finitely supported permutations of a countable …
We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some rem…
Exotic subgroups of hyperbolic groups
本文展示了双曲群的子群可具有奇异有限性性质，通过复几何与双曲几何的纤维化构造，回答了关于双曲群及其子群的旧问题。
Cyclic Subgroup Lattices as Universal Sources of Power-Type Graphs
本文建立增强幂图与有限群循环子群格的双向组合对应关系，证明二者可唯一互推，且该原理适用于其他幂型图，为有限群分析提供新框架。
For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a presentation $G=\langle A\mid R\rangle$ with a finite set of generators $A$ and a recursively enumerable set $R$ of defining relations if and only if it is a subgroup of a group $H$ finitely presented in the variety ${\cal B}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\cal B}_n$ group containing isomorphic copies of all fin…
A universal group is a subgroup of the group of type preserving automorphisms of a right-angled building and hence associated to this building. A question is then if this universal group can act chamber-transitively and with compact open stabilisers on a different right-angled building of the same type. We answer this question and define two universal groups associated to different right-angled buildings which are isomorphic as topological groups.
The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to construct a reflection subgroup of $W$ that is isomorphic to the universal Coxeter group on three generators. The results are all proved under the restriction that the Coxeter diagram of $W$ is simply l…
Universal virtual braid groups
本文引入了通用虚拟编织群UV_n(c)，提供了一种统一的代数框架，用于虚拟编织结构，证明其具有残余有限性、线性、解字和共轭问题等性质，并分类了子群分离性和Howson性质。
We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric s…
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets …
Topological Properties of Almost Abelian Groups
本文研究复殆阿贝尔李群的离散子群、紧性条件与极大紧子群等拓扑性质。
We describe here the lower garland of some lattices of intermediate subgroups in linear groups. The results are applied to the case of subgroup lattices in general and special linear groups over a class of rings, containing the group of rational points T of a maximal non-split torus in the corresponding algebraic group. It turns out that these garlands coincide with the interval of the whole lattice, consisting of subgroups between T and its normalizer.
In 2000, M. Burger and S. Mozes introduced universal groups acting on trees with a prescribed local action. We generalize this concept to groups acting on right-angled buildings. When the right-angled building is thick and irreducible of rank at least 2 and each of the local permutation groups is transitive and generated by its point stabilizers, we show that the corresponding universal group is a simple group. When the building is locally finite, these universal groups are compactly generated totally disconnected locally compact groups, and we describe the structure of the maximal compact ope…
A complete description of subgroups in the general linear group over a semilocal ring containing the group of diagonal matrices was obtained by Z.I.Borewicz and N.A.Vavilov. It is shown in the present paper that a similar description holds for the intermediate subgroups of the group of all automorphisms of the lattice of right submodules of a free finite rank R-module over a simple Artinian ring containing the group consisting of those automorphisms which leave invariant an appropriate sublattice.
Let G be a totally disconnected, locally compact group. A closed subgroup of G is locally normal if its normaliser is open in G. We begin an investigation of the structure of the family of closed locally normal subgroups of G. Modulo commensurability, this family forms a modular lattice, called the structure lattice of G. We show that G admits a canonical maximal quotient H for which the quasi-centre and the abelian locally normal subgroups are trivial. In this situation the structure lattice of H has a canonical subset called the centraliser lattice, forming a Boolean algebra whose elements c…
Every countable group that does not contain a finitely generated subgroup of exponential growth imbeds in a finitely generated group of subexponential growth. This produces in particular the first examples of groups of subexponential growth containing the additive group of the rationals.
The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric…
The Lattice of Profinite Subgroups of Protori
本文研究紧致连通阿贝尔群（protoroi）的profinite子群格，利用分解定理证明其构成同源类格，并构造万有分解，使对偶protorius的核自然涌现，从而支持有限秩无挠群作为子群的研究。
Global transfer systems and endopurity of subgroups
本文研究全局等变同伦论中的全局转移系统格，将研究范围扩展至阿贝尔群之外，证明其不可约元素受内纯子群严格控制，可从不可约元素重构整个格，相关研究发现了大量新现象并完成了计算。
It is known that a geodesic Y in an abstract reflection space X in the sense of Loos, without any assumption of differential structure, canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we prove an analog of this result stating that, if X contains an embedded hyperbolic plane H, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL(2,R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type embedded in X and can be used…
We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly bra…
Thue-Morse sequence and groups of intermediate growth
本文通过嵌入Grigorchuk群到Thue-Morse替换生成的子移位的拓扑全群中，证明了该群包含一个中间增长子群。
Stationary random subgroups in negative curvature
本研究证明Gromov双曲空间等距群的离散平稳随机子群的极限集与临界指数性质，借此解答Gelander的问题，明确谱底与万有覆叠相同的秩一局部对称空间内射半径无界。
In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(Φ,A)$, containing the elementary subgroup $E(Φ,K)$ over a subring $K\subseteq A$ provided $Φ=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice splits into a disjoint union of "sandwiches", parametrized by intermediate subrings between $K$ and $A$. In the current article a similar result is proved for $Φ=B_n$ or $C_n$, $n\ge3$, and $2=0$ in $K$. In this settings one has to introduce more sandwiches, namely, the sandwiches are parametrized by form rings $(…
We show that the automorphism group of Philip Hall's universal locally finite group has ample generics,that is, it admits comeager diagonal conjugacy classes in all dimensions.Consequently, it has the small index property, is not the union of a countable chain of non-open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall universal group. For example, we show that every generic automorphism of Hall universal group is conjugate to all of its powers, and hence has roots of all orders.
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmuller space.
Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups
本文证明 Brady 构造的双曲群非双曲有限呈示子群的 Dehn 函数具有显式多项式上界 $n^{96}$，并确定其万有覆盖 1-骨架的精确双曲性常数。
In this paper we expand on previous results, studying the extent to which one can detect fusion in certain finite groups $Γ$, from information about the universal deformation rings of absolutely irreducible $\mathbb{F}_pΓ$-modules. We consider groups which are extensions of finite irreducible subgroups of $Gl_2({\mathbb{C}})$ by elementary abelian $p$-groups of rank $2$.
Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products
本文研究了广义Furstenberg边界的刚性性质，并通过引入X-plump子群概念，拓展了关于交叉积的不可约包含和中间代数的结论。
We study actions of locally compact groups on von Neumann factors and the associated crossed-product von Neumann algebras. In the setting of totally disconnected groups we provide sufficient conditions on an action $G\curvearrowright Q$ ensuring that the inclusion $Q \subset Q \rtimes G$ is irreducible and that every intermediate subfactor is of the form $Q \rtimes H$ for a closed subgroup $H<G$. This partially generalizes a result of Izumi-Longo-Popa [ILP98] and Choda [Ch78]. We moreover show that one can not hope to use their strategy for non-discrete groups.
On Co-Maximal Subgroup Graph of a Group
本文研究群的共极大子群图，通过移除孤立顶点定义新图，并刻画其连通性等图结构性质及参数。
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We view braid groups as subgroups of braid-permutation groups. We construct a family of universal representations of braid-permutation groups, without using associators. All representations in the family are faithful, defined over $\bbQ$ by simple explicit formulae. We show that they…
This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank $n$, which is the free product $W_n$ of $n$ copies of $\mathbb{Z}/2\mathbb{Z}$. We prove that for $n\geq 5$ the natural map $\mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n))$ is an isomorphism and that every isomorphism between finite index subgroups of $\mathrm{Out}(W_n)$ is given by a conjugation by an element of $\mathrm{Out}(W_n)$.
In this article we study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. Kangaslampi and Vdovina have constructed and classified all groups acting simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. These groups gave the first examples of cocompact lattices acting simply transitively on vertices of hyperbolic triangular Kac-Moody buildings that are not right-angled. The hyperbolic buildings arise as universal covers of polyhedra glued together from with 15 triangular face…
We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. Especially, the group satisfies the analytic Novikov conjecture.
We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete classification. For small even values of n, the classification is complete up to an explicit finite number of open cases. In order to prove our main theorem, we obtain a number of other results of independent interest, notably the characterisation of the centralisers and normalisers of the finite cyclic and dicyclic subgroups of B_n(S^2), a result concerning conjugate …
Subalgebras, subgroups, and singularity
本文研究某些群的正规子群定理的非交换类比，证明其$Γ$-不变子代数的余顺从性，并利用奇异性刻画中间von Neumann子代数。
Maximal subgroups of topological full groups of shifts of finite type
该研究探讨有限型移位拓扑全群的极大子群，证明素子群胚的三分性，给出其拓扑全群正规化子为极大的判据，扩展Thompson群V相关结果并得到含交换子群的子群的极大子群数量结论。
Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over a commutative ring $R$ with $1$. We construct an infinite-dimensional group $G_V(R)$ analogous to a finite-dimensional Chevalley group over $R$. We use a $\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ and a $\mathbb{Z}$-form of an integrable highest-weight module $V$. We construct groups $G_V(\mathbb{Z})$ analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-th…
Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and degree of polynomial growth both bounded effectively. The third and fourth authors proved the optimal bound on the degree of polynomial growth of this subgroup, at the expense of making some other parts of the result ineffective. In the present paper we prove the optimal bound on the degree of polynomial growth without making any losses elsewhere. As a consequence, we show that there exist explicit positive nu…
Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure
本文研究结群的分支子群及其 profinite 和上同调结构，通过定义meridional inertia子群和全局分支子群，分析其在有限商、profinite 完成和上同调中的性质。
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