Random groups contain surface subgroups

Invariant random subgroups of the free group

Let $G$ be a locally compact group. A random closed subgroup with conjugation-invariant law is called an {\em invariant random subgroup} or IRS for short. We show that each nonabelian free group has a large "zoo" of IRS's. This contrasts with results of Stuck-Zimmer which show that there are no non-obvious IRS's of higher rank semisimple Lie groups with property (T).

Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings

Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, where E(G) is the unique maximal finite normal subgroup of G.

Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups. If $G$ is a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous $G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $p$-groups of countably infinite rank.

Infinitely presented stable groups and invariant random subgroups of metabelian groups

We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

负曲率中的平稳随机子群

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。

On rigid stabilizers and invariant random subgroups of groups of homeomorphisms

A generalization of the double commutator lemma for normal subgroups is shown for invariant random subgroups of a countable group acting faithfully on a Hausdorff space. As an application, we classify ergodic invariant random subgroups of topological full groups of Cantor minimal $\mathbb{Z}^{d}$-systems. Another corollary is that for an ergodic invariant random subgroup of a branch group, a.e. subgroup $H$ must contain derived subgroups of certain rigid stabilizers. Such results can be applied towards classification of invariant random subgroups of Grigorchuk groups.

On subnormal subgroups in division rings containing a non-abelian solvable subgroup

Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.

Random subgroups of linear groups are free

We show that on an arbitrary finitely generated non virtually solvable linear group, any two independent random walks will eventually generate a free subgroup. In fact, this will hold for an exponential number of independent random walks.

计算有限群中包含指定子群的子群数量

Counting subgroups of a finite group containing a prescribed subgroup

本文证明了有限群 $R$ 中包含指定子群 $T$ 的子群数量存在一个显式上界，该上界依赖于指数 $[R:T]$。

Faithful invariant random subgroups in acylindrically hyperbolic groups

Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group $G$ admits a weakly mixing probability measure preserving action $G \curvearrowright (X,\mathcal{B},μ)$ which is faithful but not essentially free. In other words, $G$ admits a weakly mixing nontrivial faithful IRS. We also prove that every non-elementary hyperbolic group admits a characteristic random subgroup with the same properties.

Finite permutation groups containing a regular dihedral subgroup

We present a characterization of finite permutation groups which contain a transitive dihedral subgroup.

通过平稳测度确定子群

Determining subgroups via stationary measures

研究一般度量空间等距群上随机游走，在温和条件下，证明非初等随机游走生成子群的可公度性，结果适用于特定空间，还证明双曲3 -流形基本群不同纤维子群随机游走平稳测度的奇异性。

Invariant random subgroups of groups acting on hyperbolic spaces

Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix any point of $\partial S$. We prove that every invariant random subgroup of $G$ is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of $G$). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space $(X,μ)$ either has finite stabilizers…

包含$\u27e8_{3}$型而非$\u27e8_{4}$型子群的双曲群

Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$

本研究以Italiano、Martelli和Migliorini曾研究的${\ m PO}(8,1)$非均匀格为起点，通过Dehn填充构造出含$\u27e8_{3}$型而非$\u27e8_{4}$型子群的双曲群实例。

On Invariant Random Subgroups of Block-Diagonal Limits of Symmetric Groups

We classify the ergodic invariant random subgroups of block-diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group $G$ admitting only a finite number of ergodic measures on the path-space $X$ of the associated Bratteli digram, we prove that every non-Dirac ergodic invariant random subgroup of $G$ arises as the stabil…

Neretin groups admit no non-trivial invariant random subgroups

We show that Neretin groups have no non-trivial invariant random subgroups. These groups provide first examples of non-discrete, compactly generated, locally compact groups with this property.

Invariant random subgroups of groups acting on rooted trees

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…

Groups in which every non-cyclic subgroup contains its centralizer

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with the above defined property.

Random presentations and random subgroups: a survey

This is a survey of results on random group presentations, and on random subgroups of certain fixed groups. Being a survey, this paper does not contain new results, but it offers a synthetic view of a part of this very active field of research.

Characters and invariant random subgroups of the finitary symmetric group

We will describe the relationship between the indecomposable characters of the finitary symmetric group and its ergodic invariant random subgroups; and we will interpret each Thoma character as an asymptotic limit of a naturally associated sequence of characters induced from linear characters of Young subgroups of finite symmetric groups.

平稳作用的无核原理与平稳随机子群的端

The No-Core Principle for Stationary Actions and Ends of Stationary Random Subgroups

本文证明了可数群平稳作用的无核原理，并将其应用于平稳随机子群的几何，证明了其Schreier图几乎必然有0、1、2或无穷多个端，并构造了具有任意端数的回旋镖子群。

Arithmeticity of discrete subgroups containing horospherical lattices

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.

Unimodularity of Invariant Random Subgroups

An invariant random subgroup $H \leq G$ is a random closed subgroup whose law is invariant to conjugation by all elements of $G$. When $G$ is locally compact and second countable, we show that for every invariant random subgroup $H \leq G$ there almost surely exists an invariant measure on $G/H$. Equivalently, the modular function of $H$ is almost surely equal to the modular function of $G$, restricted to $H$. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact…

$A_1$-type subgroups containing regular unipotent elements

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X = {\rm PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_1$. In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p) = (E_6,13)$ or $(E_7,19)$). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the …

Universal groups of intermediate growth and their invariant random subgroups

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.

Kesten's theorem for Invariant Random Subgroups

An invariant random subgroup of the countable group Γ is a random subgroup of Γ whose distribution is invariant under conjugation by all elements of Γ. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on Γ is strictly less than the spectral radius of the corresponding random walk on Γ/H. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approxim…

Dense Random Finitely Generated Subgroups of Lie Groups

Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.

Subgroups of simple algebraic groups containing regular tori, and irreducible representations with multiplicity 1 non-zero weights

Our main goal is to determine, under certain restrictions, the maximal closed connected subgroups of simple algebraic groups containing a regular torus. We call a torus regular if its centralizer is abelian. We also obtain some results of independent interest. In particular, we determine the irreducible representations of simple algebraic groups whose non-zero weights occur with multiplicity 1.

Out(F_N)中托雷利子群的刚性

Rigidity of the Torelli subgroup in $Out(F_N)$

本文证明了N≥4时Out(F_N)中托雷利子群等多类子群的单同态刚性（仅差共轭），并将结论推广到N=3的部分子群，还得出其抽象 commensurator性质及co-Hopfian性等应用。

逐元素保守动作与新的鼓浪子群构造

Elementwise conservative actions and new constructions of boomerang subgroups

本文研究了自由群中逐元素保守随机子群的构造，证明其在可测满群中的普遍性，并补充了高维晶格中相关刚性结果。

Amenable Invariant Random Subgroups

We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology.

Co-induction and Invariant Random Subgroups

In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of $\mathbb{F}_2$ which are all invariant and weakly mixing with respect to the action of $\text{Aut}(\mat…

On surface subgroups of doubles of free groups

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.

紧不变随机子群

Compact Invariant Random Subgroups

研究了能完全测度紧子群的遍历不变随机子群，证明在实李群、紧p-进李群、局部紧双曲群和无限端群中，此类子群总包含于紧正规子群中；在一般p-进李群中，包含于局部椭圆根；在完全不连通局部紧群中，包含于所有内自同构李子群交集。

A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated

Assume that all the Sylow subgroups of a finite group $G$ can be generated by $d$ elements. Then the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found, is at most $d+η$ with $η\sim 2.875065.$

双曲反射群中的不变随机子群

Invariant random subgroups in hyperbolic reflection groups

本文证明了双曲反射群存在具有不可数多种子群同构类型的不变随机子群，并通过Coxeter多面体构造回答了S.Thomas等人的问题。

The quaternion group as a subgroup of the sphere braid groups

Let n be greater than or equal to 3. We prove that the quaternion group of order 8 is realised as a subgroup of the sphere braid group B\_n(S^2) if and only if n is even. If n is divisible by 4 then the commutator subgroup of B\_n(S^2) contains such a subgroup. Further, for all n greater than or equal to 3, B\_n(S^2) contains a subgroup isomorphic to the dicyclic group of order 4n.

Random generation of finitely generated subgroups of a free group

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has …

某些轨道上的不变随机子群

Invariant random subgroups on certain orbits

本文研究了连通李群在共轭作用下某些轨道上不变随机子群的存在性问题。

自由拟凸子群的严格上升链

Ascending Chains of Free Quasiconvex Subgroups

本文证明了双曲群中不存在常秩自由拟凸子群的严格上升链，这一结果深化了对双曲群子群结构的理解。

Invariant random subgroups of semidirect products

We study invariant random subgroups (IRSs) of semidirect products $G = A \rtimes Γ$. In particular, we characterize all IRSs of parabolic subgroups of $\mathrm{SL}_d(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^d \rtimes \mathrm{SL}_d(\mathbb{R})$ are either of the form $\mathbb{R}^d \rtimes K$ for some IRS of $\mathrm{SL}_d(\mathbb{R})$, or are induced from IRSs of $Λ\rtimes \mathrm{SL}(Λ)$, where $Λ< \mathbb{R}^d$ is a lattice.

分支群的随机子群

Random subgroups of branch groups

本文证明超强分形分支投射有限群中的独立 Haar 随机元素生成在树边界上自由作用的自由子群，改进了 Abért 的弱分支群结果并推广了迭代圈积情形。

子群图包含大度数顶点的有限群

Finite groups whose subgroup graph contains a vertex of large degree

本文研究子群图中顶点度数超过 |G|/2-1 的有限群，并对有限可解群给出完整分类，推广了 Burness 和 Scott 关于素数阶子群个数的结果。

On profinite groups in which commutators are covered by finitely many subgroups

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…

On normal subgroups of the braided Thompson groups

We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such normal subgroups. For Fbr, we prove that for any normal subgroup N of Fbr, either N is contained in the kernel of Fbr \onto F, or else N contains [Fbr,Fbr]. We also compute the Bieri-Neumann-Strebel invariant Sigma^1(Fbr), which is a useful tool for understanding normal subgroups containing the commutator subgroup.

The profinite completion of a group localised at a subgroup

Let $G$ be a group and let $K$ be a commensurated subgroup of $G$. Then there is a totally disconnected, locally compact (t.d.l.c.) group $\hat{G}_K$ that contains the profinite completion of $K$ as an open compact subgroup and also contains $G$ (modulo the finite residual of $K$) as a dense subgroup. Moreover, given an arbitrary group $G$, then every t.d.l.c. group containing an image of $G$ as a dense subgroup can be realised as a quotient of $\hat{G}_K$ for some commensurated subgroup $K$.

Combinatorial Methods for Detecting Surface Subgroups in Right-Angled Artin Groups

We give a short proof of the following theorem of Sang-hyun Kim: if $A(Γ)$ is a right-angled Artin group with defining graph $Γ$, then $A(Γ)$ contains a hyperbolic surface subgroup if $Γ$ contains an induced subgraph $\bar{C}_n$ for some $n \geq 5$, where $\bar{C}_n$ denotes the complement graph of an $n$-cycle. Furthermore, we give a new proof of Kim's co-contraction theorem.

Classification of Sylow classes of parabolic and reflection subgroups in unitary reflection groups

Let $\ell$ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an $\ell$-Sylow subgroup. The classification assists in describing the $\ell$-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an $\ell$-Sylow subgroup is $G$ itself, the reflection subgroup within th…

Classification of subgroups of symplectic groups over finite fields containing a transvection

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations.

Left ordered groups with no nonabelian free subgroups

There has been interest recently concerning when a left ordered group is locally indicable. Bergman and Tararin have shown that not all left ordered groups are locally indicable, but all known examples contain a nonabelian free subgroup. We shall show for a large class of groups not containing a nonabelian free subgroup, that any left ordered group in this class is locally indicable. Specifically this class is the smallest class of groups containing NS and Thompson's group of piecewise linear homeomorphisms of the unit interval, and is closed under taking subgroups, quotient groups, extensions…

Lower central subgroups of a free group and its subgroup

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does not normally generate $F$. The question comes from a study of Hirzebruch-type invariants from iterated $p$-covers for 3-dimensional homology cylinders.

A Polish group containing a Haar null $F_σ$-subgroup that cannot be enlarged to a Haar null $G_δ$-set

Answering a question of Elekes and Vidnyánszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_σ$-set in $H$ that cannot be enlarged to a Haar null $G_δ$-set.

Character Theory of Symmetric Groups, Subgroup Growth of Fuchsian Groups, and Random Walks

We derive an asymptotic expansion for the subgroup of arbitrary Fuchsian groups and some other classes of large groups. Moreover, the main conjecture for Random Walks on symmetric groups is established in full generality. Both problems require subtle estimates of character values and root number multiplicities for symmetric groups, which are also of independent interest.

Invariant random subgroups of the lamplighter group

Let $G$ be one of the lamplighter groups $({\mathbb{Z}/p\bz})^n\wr\mathbb{Z}$ and $\Sub(G)$ the space of all subgroups of $G$. We determine the perfect kernel and Cantor-Bendixson rank of $\Sub(G)$. The space of all conjugation-invariant Borel probability measures on $\Sub(G)$ is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If $F$ is a finite group and $Γ$ an infinite group which does not have property $(T)$ then the conjugation-invariant probability measures on $\Sub(F\wrΓ)$ supported on $\oplus_ΓF$ als…