On co-Hopfian nilpotent groups

关于不可数Hopfian与co-Hopfian阿贝尔群的存在性

On the existence of uncountable Hopfian and co-Hopfian abelian groups

本文研究不可数co-Hopfian与绝对Hopfian阿贝尔群的存在性，证明了在特定基数条件下不存在某些co-Hopfian群，并构造了任意基数的绝对Hopfian无挠阿贝尔群。

Non-hopfian relatively free groups

To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960's, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-hopfian.

关于强Hopfian混合阿贝尔群

On Strongly Hopfian Mixed Abelian Groups

本文研究混合阿贝尔群中（一致）强Hopfian性和（一致）强co-Hopfian性，证明约化强co-Hopfian群必为强Hopfian，并将无扭群的特征推广到Warfield群。

阿贝尔群与幂零群中刚性条件的反分类结果

Anti-Classification Results for Rigidity Conditions in Abelian and Nilpotent Groups

本文证明了刚性阿贝尔群、Hopfian群及co-Hopfian群在相应可数群空间中的完备共解析性，并解决了Thomas提出的两个公开问题，揭示了四类刚性条件的最高复杂度。

通过剩余有限性与Hopf性质探讨共轭quandle及其底层群之间的联系

Connections between conjugation quandles and their underlying groups via residual finiteness and the Hopf property

本文研究了共轭quandle与其底层群的Hopf性质和剩余有限性联系，证明了Hopf性质的传递性并提供反例，同时利用Dehn quandle完全刻画了Baumslag-Solitar群共轭quandle的剩余有限性。

Cofinitely Hopfian groups, open mappings and knot complements

A group $Γ$ is defined to be cofinitely Hopfian if every homomorphism $Γ\toΓ$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented.

Co-Hopfian virtually free groups and elementary equivalence

We prove that two co-Hopfian finitely generated virtually free groups are elementarily equivalent if and only if they are isomorphic. We also prove that co-Hopfian finitely generated virtually free groups are homogeneous in the sense of model theory.

强且一致强的余霍普夫群

Strongly and Uniformly Strongly co-Hopfian Abelian Groups

本文研究了强且一致强余霍普夫群的结构，证明了其与可除群和sp群的关系，并探讨了其在代数紧致性和cotorsion性质中的应用。

A non-Hopfian almost convex group

In this article we prove that an "isometric multiple HNN-extension" of a group satisfying the falsification by fellow traveler property is almost convex. As a corollary, Wise's example of a CAT(0) non-Hopfian group is Almost convex.

co-Hopfian阿贝尔群的两个推广

Two Generalizations of co-Hopfian Abelian Groups

本文定义了广义co-Hopfian群和相对co-Hopfian群两类，完全刻画了p-群、无挠群及混合群的相应性质，并证明相对co-Hopfian p-群恰为co-Hopfian群，且两类遗传群在秩意义下重合。

The group of unimodular automorphisms of $\mathbb{C}^2$ is hopfian

Let $G$ be the group of unimodular automorphisms of $\mathbb C^2$. In the paper we prove two interesting results about this group. The first one is about absence of non-trivial finite-dimensional representations of $G$. The second one, we show that any non-trivial group endomorphism of $G$ is a monomorphism, which implies that $G$ is hopfian.

Hopfian阿贝尔群的两种推广

Two Generalizations of Hopfian Abelian Groups

本文通过定义相对Hopfian群和弱Hopfian群推广交换情形的Hopfian群概念，刻画了约化阿贝尔p-群与无挠群中两类Hopf性的一致与差异，并构造出非Hopfian的相对Hopfian群。

Finitely presentable, non-Hopfian groups with Kazhdan's Property (T) and infinite outer automorphism group

We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.

余霍普夫性不是 profinite 性质

Co-Hopfianity is not a profinite property

本文研究了两个有限生成余有限群G和H，其 profinite 完成同构，但G是余霍普夫群而H不是。通过Wise的余有限Rips构造，构造了具有平凡 profinite 完成的无向群U，并利用H作为G的预像子群，其共轭于自身真子群，从而得出H非余霍普夫性。

关于余霍普夫群和环的注记

A note on co-Hopfian groups and rings

本文证明当k是2元域的代数闭包时，SL₂(k)是余霍普夫群，并扩展了Bunina和Kunyavskii的结果，同时证明满足Turner退化定理的群类不闭合于初等等价下。

A family of two generator non-Hopfian groups

We construct $2$-generator non-Hopfian groups $G_m, m=3, 4, 5, \dots$, where each $G_m$ has a specific presentation $G_m=\langle a, b \, | \, u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}= \cdots =1 \rangle$ which satisfies small cancellation conditions $C(4)$ and $T(4)$. Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group of slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)\langle m \rangle,m+1,m]$ in continued fraction expansion for every integer $i \ge 1$.

Structure of normally and finitely non-co-Hopfian groups

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to G. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: First, we show any characteristic proper finite-index subgroup isomorphic to G arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (fo…

关于阿贝尔群的几种霍普菲性质版本

On Some Versions of Hopficity for Abelian Groups

本文研究了阿贝尔群的几种霍普菲性质版本，包括余有限霍普菲群和几乎余有限霍普菲群，探讨了其定义和性质。

无穷逻辑的表达能力与绝对余Hopf性质

Expressive Power of Infinitary Logic and Absolute co-Hopfianity

针对绝对余Hopf群，本文利用无穷逻辑证明第一个优美基数之上不存在绝对余Hopf阿贝尔群，且将结果推广到交换环模范畴，解决了其存在性与大小限制问题。

The Hopfian Property of $n$-Periodic Products of Groups

Let $H$ be a subgroup of a group $G$. A normal subgroup $N_H$ of $H$ is said to be inheritably normal if there is a normal subgroup $N_G$ of $G$ such that $N_H=N_G\cap H$. It is proved in the paper that a subgroup $N_{G_i}$ of a factor $G_i$ of the $n$-periodic product $\prod_{i\in I}^nG_i$ with nontrivial factors $G_i$ is an inheritably normal subgroup if and only if $N_{G_i}$ contains the subgroup $G_i^n$. It is also proved that for odd $n\ge 665$ every nontrivial normal subgroup in a given $n$-periodic product $G=\prod_{i\in I}^nG_i$ contains the subgroup $G^n$. It follows that almost all $…

可逆的组合缠绕积

Hopfian combinatorial wreath products

该研究探讨了组合缠绕积的可逆性条件，通过构造具体例子展示了不可逆情况，并分析了自动群的结构。

XL型阿廷群之间的同态

Homomorphisms between XL-type Artin groups

该文研究XL型阿廷群的同态性质，证明一般阿廷群兼具霍普夫性与余霍普夫性，刻画了标记≥6的一般图阿廷群的余霍普夫性及外自同构群有限性的条件，还给出无割点时自同构群的生成元集。

阿贝尔群的完备化的Hopficity

Hopficity of profinite completions of abelian groups

研究阿贝尔群完备化的拓扑Hopficity问题，通过证明\(\widehat{A}\)是拓扑Hopfian当且仅当对每个素数\(p\)，\(A/pA\)有限，否定回答了Kourovka笔记本中的问题6.30。

Co-Hopfian 与有界内自同构刚性的混合阿贝尔群

Co-Hopfian and boundedly endo-rigid mixed abelian groups

该研究在温和基数条件下构造了指定挠子群的有界内自同构刚性阿贝尔群，给出配对完整刻画，应用双重黑箱方法，并解决了大基数下co-Hopfian混合阿贝尔群的存在性问题并给出判定准则。

自由阿贝尔乘曲面群中自同态的不动子群分类

Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups

本文研究自由阿贝尔乘曲面群等群类的自同态性质，刻画其自同态双射条件，证明相关群的Hopf性，完成自同态不动子群完整分类并给出有限不动子群的等价条件。

Automorphisms of the Baumslag-Gersten group

We classify homomorphisms of the Baumslag-Gersten group into itself. We prove it is Hopfian and co-Hopfian. We show that the group of outer automorphisms of the Baumslag-Gersten group is isomorphic to the dyadic rationals with the addition operation. These results are not new. They were obtained by Andrew M. Brunner in \cite{Brunner}. However, our exposition is self-contained and, hopefully, more accessible for some readers.

Hopfian 环积与稳定有限性猜想

Hopfian wreath products and the stable finiteness conjecture

本文研究阿贝尔基群环积的 Hopf 性质，证明 Kaplansky 稳定有限性猜想等价于所有有限生成阿贝尔群与有限生成 Hopfian 群的环积均为 Hopfian，并进一步将其与加法细胞自动机的 Gottschalk 满射性猜想联系起来。

Endomorphisms of relatively hyperbolic groups

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. (2) If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic group is not Hopfian, then H acts non-trivially on an R-tree. (3) Every non-elementary relatively hyperbolic group has a non-elementary relative…

Automorphisms and endomorphisms of lacunary hyperbolic groups

In this article we study automorphisms and endomorphisms of lacunary hyperbolic groups. We prove that every lacunary hyperbolic group is Hopfian, answering a question by Henry Wilton. In addition, we show that if a lacunary hyperbolic group has the fix point property for actions on $\mathbf R$-trees, then it is co-Hopfian and its outer automorphism group is locally finite. We also construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance.

Le groupe de Cremona est hopfien

We describe the endomorphisms of the Cremona group and obtain that the Cremona group is hopfian.

Non-residually finite groups hyperbolic relative to residually finite subgroups

Let $m$ and $k$ be integers such that $|m|, \, |k| >1$ and $\gcd (m,k)=1$. We show that all Baumslag-Solitar groups $BS(m,mk)$ are non-residually finite groups hyperbolic relative to residually finite subgroups. By a result of Osin (2007), this implies that there exists a non-residually finite hyperbolic group, thus solving a long-standing open problem of Gromov (1987). We also show that all $BS(m, mk)$ are non-Hopfian groups hyperbolic relative to Hopfian subgroups.

Quotients and subgroups of Baumslag-Solitar groups

We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).

Averaged Dehn Functions for Nilpotent Groups

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl^α$ for $α>2$ then it satisfies the averaged isoperimetric inequality $δ^{\text{avg}}(l)<C'l^{α/2}$. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.

Scaled relators and Dehn functions for nilpotent groups

A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. Using this method, we give new proofs of some results on Dehn functions of nilpotent groups, prove theorems on central powers and certain quotients of nilpotent groups, and construct the first example of …

关于直有限 Abel 群的一些遗传类与超类

On Some Hereditary and Super Classes of Directly Finite Abelian Groups

本文研究直有限 Abel 群相关类的遗传与超性质，证明相对 Hopf 群具备相应性质当且仅当它是扩张 Bassian 的，并给出更一般背景下的充要条件。

自由斜辫子与Yang-Baxter方程的自由解

Free Skew Braces and Free Solutions of the Yang--Baxter Equation

证明Yang-Baxter方程的自由非退化双射解的结构斜辫子是自由的，并构造任意类的自由右幂零斜辫子，证明其可解词问题及有限解中的律对一切解成立。

群的内自同构的改进刻画

An improved characterisation of inner automorphisms of groups

本文证明任意群G可嵌入到一个简单、完备的co-Hopfian群H中，非平凡自同态仅在为内自同构时可延伸到H，改进了Schupp的定理。

Bassian-有限阿贝尔群

Bassian-Finite Abelian Groups

引入严格位于co-Hopfian群与Dedekind-finite群之间的Bassian-finite阿贝尔群新类，证明其在挠群情况下与co-Hopficity等价，并构造了无挠非co-Hopfian及非Bassian-finite的Dedekind-finite群实例。

Limits of (certain) CAT(0) groups, II: The Hopf property and the shortening argument

This is the second in a series of papers about torsion-free groups which act properly and cocompactly on a CAT(0) metric space with isolated flats and relatively thin triangles. Our approach is to adapt the methods of Sela and others for word-hyperbolic groups to this context of non-positive curvature. The main result in this paper is that (under certain technical hypotheses) such a group as above is Hopfian. This (mostly) answers a question of Sela.

微分同胚germ的代数性质

Algebraic properties of the group of germs of diffeomorphisms

研究微分同胚germ群Diff(C^n,0)及其形式完备化群的代数性质，证明其交换子、余有限性及Hopfian性质，探讨嵌入该群的群结构约束。

Abelian Groups, Homomorphisms and Central Automorphisms of Nilpotent Groups

In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.

Residually Torsion-Free Nilpotent One Relator Groups

We show that certain cyclically pinched one-relator groups are residually torsion-free nilpotent.

Palindromic width of free nilpotent groups

In this paper we consider the palindromic width of free nilpotent groups. In particular, we prove that the palindromic width of a finitely generated free nilpotent group is finite. We also prove that the palindromic width of a free abelian-by-nilpotent group is finite.

The nilpotent genus of finitely generated residually nilpotent groups

If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there exists a monomorphism of $G$ into $H$ which induces isomorphisms between the corresponding quotients of their lower central series. We first consider residually nilpotent groups and find sufficient conditions on the monomorphism so that $H$ is para-$G.$ We then prove that for certain polycyclic groups, if $H$ is para-$G$, then $G$ and $H$ have the same Hirsch l…

On the geometry of free nilpotent groups

In this article, we study geometric properties of nilpotent groups. We find a geometric criterion for the word problem for the finitely generated free nilpotent groups. By geometric criterion, we mean a way to determine whether two words represent the same element in a free nilpotent group of rank $r$ and class $k$ by analyzing their behavior on the Cayley graph of the free nilpotent group of rank $r$ and class $k-1$.

Fitting quotients of finitely presented abelian-by-nilpotent groups

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.

The subgroup identification problem for finitely presented groups

We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as an investigation into the difference between strong and weak effective coherence for finitely presented groups.

Maximal abelian normal subgroups in Nilpotent groups of class 2

A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free nilpotent group of class 2 such a group is always a cyclic extension of the center. However this need not be true in an arbitrary finitely generated class two nilpotent group. We show that three generator nilpotent groups violating this condition can be easily found. However in the case when there are four or more generators certain conditions must hold.

Relatively free nilpotent torsion-free groups and their Lie algebras

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relativ…

自由幂零群的完全外自同构群

Complete outer automorphism groups of free nilpotent groups

本研究证明了2类无限生成自由幂零群的外自同构群具有完全性，推进了无限生成自由幂零群外自同构群的结构性质研究。

On ${\cal N}_c$-Covering Groups of a Nilpotent Product of Cyclic Groups

The first author, in 2001, presented a structure for the Baer-invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups. In this paper, using the above structure, we prove the existence of an ${\cal N}_c$-covering group for a nilpotent product of a family of cyclic groups. We also present a structure for the ${\cal N}_c$-covering group.

Locally pro-p contraction groups are nilpotent

The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent. We show here that the torsion factor is nilpotent too, and hence that every locally pro-p contraction group is nilpotent.

Primitive finite nilpotent linear groups over number fields

Building upon the author's previous work on primitivity testing of finite nilpotent linear groups over fields of characteristic zero, we describe precisely those finite nilpotent groups which arise as primitive linear groups over a given number field. Our description is based on arithmetic conditions involving invariants of the field.