Closure Properties in the Class of Multiple Context Free Groups

Fixed subgroups and computation of auto-fixed closures in free-abelian times free groups

The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of $m,n$. We also study periodic points of endomorphisms of $Z^m\times F_n$, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of $Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real eleme…

关于自由群的循环子群在pro-V拓扑中的闭包

On the closure of cyclic subgroups of a free group in pro-V topologies

本文研究自由群的循环子群在各类pro-V拓扑下的闭包问题，证明其在pro-幂零拓扑下总是闭的，并具体计算了pro-$\mathbf{G}_p$和pro-$\mathbf{V}_p$拓扑下的闭包。

The pro-nilpotent group topology on a free group

In this paper, we work on the pro-nilpotent group topology of a free group. First we investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group topology. We present an algorithm for the calculation of the closure in the pro-nilpotent group topology of the product of finitely many finitely generated subgroups of a free group. Then we deduce that the nil-closure of a rational subset is computable. We also prove that the pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V, and J * G_nil are decidable.

Zariski Closures and Subgroup Separability

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group.

带受限非交换合成因子的置换群的闭包

Closures of permutation groups with restricted nonabelian composition factors

该研究针对计算群论相关问题，证明了当$k\geq4$且$d\geq25$时，$\text{Alt}(d)$-自由置换群的$k$-闭包仍保持$\text{Alt}(d)$-自由的性质。

带系数群的内射性定理和代数闭包

Injectivity theorems and algebraic closures of groups with coefficients

最近，Cochran和Harvey定义了群的无挠导出列，并证明了相应的无挠导子的内射性定理。我们表明，有一个普遍的建设，扩展这样的内射定理的同构定理。我们的结果涉及到一定的同调局部化群的内射性定理。为了给出必要同调局部化的具体组合描述和存在性证明，我们通过考虑某类方程，引入了系数群的一种新的代数闭包。

Subnormal subgroups in free groups, their growth and cogrowth

In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (3) gives sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal closures of free generators in F.

自由群中的代数扩张

Algebraic extensions in free groups

本文的目的是统一最近三篇独立论文（Ventura 1997、Margolis、Sapir和Weil 2001以及Kapovich和Miasnikov 2002）的观点，其中给出了Takahasi 1951年定理的类似现代版本。我们开发了一个理论的代数扩张自由群，突出的类比和差异，相对于相应的经典领域理论的概念，我们详细讨论的概念，代数闭包。我们应用这一理论的研究和计算的某些代数性质的子群（例如异常，纯，惰性或压缩，被关闭在某些profinite拓扑）和相应的封闭运营商。我们还分析了封闭的子群下的某些方程组的解决方案。

The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the mth power of a Dehn twist has infinite order, and for some integers m the quotient contains a free nonabelian subgrou…

自由群上的pro-$k$-可解拓扑

The pro-$k$-solvable topology on a free group

本文证明了自由群有限生成子群在pro-(元)交换和pro-${\bf V}$拓扑下闭包性质的可判定性，并给出了闭包基的有效构造及成员资格判定。

R. Thompson群V的字问题的复杂性

On the complexity of the word problem of the R. Thompson group V

本研究分析Lehnert与Schweitzer的证明，得出R. Thompson群V及其有限生成子群的字问题的复杂性，确定其属于对数DCFL且在确定性多带图灵机上具二次时间复杂度。

自由群中的充足性

Ampleness in the free group

我们表明，理论的自由群-更一般的理论，任何挠自由双曲群-是$n$ -任何$n\geq 1$的例子。我们也给出了自由群中虚代数闭包的一个明确的描述。

Invariant laminations for irreducible automorphisms of free groups

For every atoroidal iwip automorphism $ϕ$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(ϕ)$ is obtained as "diagonal closure" of the support of the backward limit current $μ_-(ϕ)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(ϕ)$, in terms of Bestvina-Fe…

On the structure of some contranormal-free groups

A subgroup of a group is contranormal if its normal closure coincides with the group. We call such groups without proper contranormal subgroups contranormal-free. In this paper we prove various results concerning contranormal-free groups proving, for example that locally generalized radical contranormal-free groups which have finite section rank are hypercentral.

Free products from spinning and rotating families

The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten the proof of the Dahmani-Guirardel-Osin theorem.