Martin R. Bridson, James Howie
If S is a subgroup of a direct product of two limit groups, and S is of type FP(2) over the rationals, then S has a subgroup of finite index that is a direct product of at most two limit groups.
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We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.
If $G_1,...,G_n$ are limit groups and $S\subset G_1\times...\times G_n$ is of type $\FP_n(\mathbb Q)$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.
In this paper, we give some necessary and sufficient conditions for a normal subgroup of an amalgamated product of groups to be finitely generated. We apply these conditions together with Stallings' fibering theorem to prove that an irreducible multilink in a homology 3-sphere fibers if and only if each of its multilink splice components fibers.
We exploit Zlil Sela's description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if $G_1,...,G_n$ are in E then a subgroup $Γ\subset G_1\times...\times G_n$ is of type $\FP_n$ if and only if $Γ$ is itself, up to finite index, the direct product of at most $n$ groups from $\mathcal E$. This answers a question of Sela.
A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, o…
We partly generalize the estimate for the rank of intersection of subgroups in free products of groups, proved earlier by S.V.Ivanov and W.Dicks, to the case of free amalgamated products of groups with normal finite amalgamated subgroup. We also prove that the obtained estimate is sharp and cannot be further improved when the amalgamated product contains an involution.
Dehn functions of subgroups of products of free groups. Part II: Precise computations
本文精确计算了 Bridson-Dison 群的 Dehn 函数为四次，并证明自由群直积中满足特定有限性型和余秩条件的余阿贝尔子群具有二次 Dehn 函数。
Dehn functions of subgroups of products of free groups. Part I: Uniform upper bounds
本文证明三个自由群直积的有限展示子群及 n 个自由群直积中 $\mathcal{F}_{n-1}$ 型子群的 Dehn 函数均以 $N^9$ 为上界，正面回答 Dison 问题并为 Bridson 猜想提供新证据。
It is proved that all finitely generated subgroups of generalized free product of two groups are finitely separable provided that free factors have this property and amalgamated subgroups are normal in corresponding factors and satisfy the maximum condition for subgroups.
Klein-Maskit combination theorem for Anosov subgroups: Free products
本文在自由积情形推广 Klein-Maskit 组合定理，证明非紧型半单 Lie 群中两个满足拓扑条件的 Anosov 子群生成的群仍为 Anosov 且同构于其自由积。
This paper gives necessary and sufficient conditions that the free product with amalgamation of circularly-ordered groups admit a circular ordering extending the given orderings of the factors. Our result follows from establishing a categorical framework that allows the problem to be restated in terms of amalgamating certain left-ordered central extensions, where we are able to apply work of Bludov and Glass.
Kurosh Subgroup Theorem in Free Pro-p Products
本文研究了自由p-级数积中闭子群的结构，提出一个充分条件以确定其为自由p-级数积。
On subgroup growth of iterated wreath products in product action
本文研究了通过迭代wreath产物在积作用下得到的hereditarily just infinite profinite群的子群增长类型，证明了存在任意子群增长类型的群。
We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of certain type do occur as closed subgroups of free profinite products of two profinite groups. This also solves a question of Pop about prosolvable subgroups of free profinite products.
This paper is devoted to the investigation of the property of order separability for HNN extensions and free products with commutative subgroups. Particularly it was proven that HNN extension of a free group with maximal connected cyclic subgroups is 2-order separable.
A conjecture of Dicks and the author on rank of the intersection of factor-free subgroups in free products of groups is proved for the case of left ordered groups.
We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.
For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) to be verbally closed.
Finite subgroups of automorphisms of free products
本文研究自由积外自同构的有限子群，给出了其阶的上界及特定条件下挠外自同构的阶的界。
Cartesian subgroups in graph products of groups
该研究推广了直角Coxeter群与Artin群的换位子群构造，借助多面体积理论给出群图积的笛卡尔子群关系数及亏格的上下界，还提出计算其小型表现的算法。
We give an explicit characterization of which direct products $G$ of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of $G$ is compressed, and of which is it always inert.
We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.
We study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated)…
An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characterictic.
Irreducible discrete subgroups in products of simple Lie groups
该研究构造了$SL(2,\mathbb{R})\times SL(2,\mathbb{R})$中一个非格的不可约离散子群，解答了文献[15]提出的问题。
We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products. Furthermore, we develop the tools to study the analogous question in the pro-$p$ case. For a wide class of groups we show that the relevant cyclic subgroups - which are called $p$-isolated - are closed in the pro-$p$ topology of the graph product. In particular, we show that every $p$-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-$p$ topology, and we fully characterise such subgroups.
We construct torsion-free hyperbolic groups without unique product whose subgroups up to some given finite index are themselves non-unique product groups. This is achieved by generalising a construction of Comerford to graphical small cancellation presentations, showing that for every subgroup $H$ of a graphical small cancellation group there exists a free group $F$ such that $H*F$ admits a graphical small cancellation presentation.
We show that the product of two partial normal subgroups of a locality (in the sense of Chermak) is again a partial normal subgroup. This generalizes a theorem of Chermak and fits into the context of building a local theory of localities.
Let G be the free product of groups A and B with commuting subgroups H \leqslant A and K \leqslant B, and let C be the class of all finite groups or the class of all finite p-groups. We derive the description of all C-separable cyclic subgroups of G provided this group is residually a C-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p'-isolated in the free factors, then all p'-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p'-isolated and cyclic.
We show that the class of amalgamated free products of two free groups over a cyclic subgroup admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.
In this paper we compute the subgroup zeta functions of nilpotent semi-direct products of groups of the form $$G_n=<x_1,...,x_n,y_1,...,y_{n-1}|[x_i,x_n]=y_i, 1\leq i\leq n-1, \text{all other $[,]$ trivial}>$$ and deduce local functional equations.
Stable subgroups of graph products
研究图积中稳定子群的性质，证明其等价于在关联接触图中准等距嵌入的子群，并推广了Koberda等人关于稳定与纯洛希多里子群等价的结果。
We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche-Wendt groups.
In the context of group-theoretic fast matrix multiplication the TPP capacity is used to bound the exponent $ω$ of matrix multiplication. We prove a new and sharper upper bound for the TPP subgroup capacity of a finite group
In this paper, we describe the automorphism group of semidirect product of two groups that fixes the non-normal subgroup of it. We have computed these automorphisms for the non-abelian metacyclic $p$-group and non-abelian $p$-groups $(p\ge 5)$ of order $p^{4}$, where $p$ is a prime.
Let $G$ be a free product of two groups with amalgamated subgroup, $π$ be either the set of all prime numbers or the one-element set \{$p$\} for some prime number $p$. Denote by $Σ$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $π$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $π$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $π^{\prime}$-isolated, don't belong to $Σ$. Some sufficient conditions are obtaine…
We determine all maximal subgroups of the direct product $\sc G^n$ of $\sc n$ copies of a group~$\sc G$. If $\sc G$ is finite, we show that the number of maximal subgroups of~$\sc G^n$ is a quadratic function of~$\sc n$ if $\sc G$ is perfect, but grows exponentially otherwise. We~deduce a theorem of Wiegold about the growth behaviour of the number of generators of~$\sc G^n$.
Finitely presented subgroups of direct products of graphs of groups with free abelian vertex groups
本文将自由群直积有限表现子群的相关结构定理推广到二维凝聚右角阿廷群（RAAGs）直积，刻画了其有限表现及$FP_n$型子群的结构，并证明这类子群的多重共轭问题与成员问题可判定。
Dehn functions of coabelian subgroups of direct products of groups
本研究提出计算群直积的余阿贝尔子群精确Dehn函数的新方法，推广了前人相关结果，并将其应用于多类群的相关研究中。
Let $G$ be a group. We give an explicit description of the set of elements $x \in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H \leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?
Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.
Let $G=\ast_{i=1}^{n}G_{i}$ and let $ϕ$ be a symmetric endomorphism of $G$. If $ϕ$ is a monomorphism or if $G$ is a finitely generated residually finite group, then the fixed subgroup $Fix(ϕ)=\{g\in G:ϕ(g)=g\}$ of $ϕ$ has Kurosh rank at most $n$.
We derive a generating series for the number of free subgroups of finite index in $Δ^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $Δ^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $Δ^+$, which correspond to isomorphism classes of hypermaps. Asymptotic f…
A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of `Cartesian decompositions' of the permuted set, relating them to certain `Cartesian systemsof subgroups'. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.
Exponentially distorted subgroups in wreath products
本文证明在两类圈积群 $\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$ 与 $\mathbb{Z} \wr F_2$ 中均存在指数扭曲的子群。
Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is shown to be decidable. An algorithm for this problem is constructed and its complexity is analysed. Groups in this class are also shown to possess the Howson property.
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.
It is proved that every group of the form $G=AB$ with two subgroups $A$ and $B$ each of which is either abelian or has a quasicyclic subgroup of index $2$ is soluble of derived length at most $3$. In particular, if $A$ is abelian and $B$ is a locally quaternion group, this gives a positive answer to Question 18.95 of "Kourovka notebook" posed by A.I.Sozutov.
We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.
The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the commutator width of such groups is 1 and we research some properties of its commutator subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and …
For a non-cyclic finite group $X$ let $σ(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $σ(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$.
Let $G_1 \times G_2$ be a subgroup of $\mathrm{SO}_3(\mathbb{R})$ such that the two factors $G_1$ and $G_2$ are non-trivial groups. We show that if $G_1 \times G_2$ is not abelian, then one factor is the (abelian) group of order 2, and the other factor is non-abelian and contains an element of order 2. There exist finite and infinite such non-abelian subgroups.
We prove that the reduced Kurosh rank of the intersection of two subgroups $H$ and $K$ of a free product of right-orderable groups is bounded above by the product of the reduced Kurosh ranks of $H$ and $K$. In particular, taking the fundamental group of a graph of groups with trivial vertex and edge groups, and its Bass-Serre tree, our Theorem becomes the desired inequality of the usual Strengthened Hanna Neumann conjecture for free groups.
In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a $Δ$-product or $\ast$-product from the values of its corresponding supercharacters.
For a non-empty class of groups $\cal L$, a finite group $G = AB$ is said to be an $\cal L$-connected product of the subgroups $A$ and $B$ if $\langle a, b\rangle \in \cal L$ for all $a \in A$ and $b \in B$. In a previous paper, we prove that for such a product, when $\cal L = \cal S$ is the class of finite soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In parti…
Subgroups of the direct product of graphs of groups with free abelian vertex groups
本文将 Baumslag–Roseblade 关于自由群直积有限展示子群的结果推广到具有自由阿贝尔顶点群与循环边群的群图，并证明相关 RAAG 子群的算法问题可判定。
We determine the asymptotic number of index $n$ subgroups in virtually cyclic right-angled Coxeter groups and their free products as $n\to\infty$.
A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy proble…
The object of this paper is to find a necessary and sufficient condition for the groups $G_1, G_2, ..., G_n$ so that every normal subgroup of the product $\prod_{i=1}^{n} G_i$ is of the type $\prod_{i=1}^{n} N_i$ with $N_i \trianglelefteq G_i$, $i=1,2, ..., n$. As a consequence we obtain a well-known result due to R. Remak about centreless completely reducible groups having finitely many direct factors.
An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups. However, not much is yet learned about the structure of arbitrary quasiconvex subsets. In this work we study the properties of products of quasiconvex subgroups; we show that such sets are quasiconvex, their finite intersections have a similar algebraic representation and, thus, are quasiconvex too.
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