S. V. Ivanov, A. M. Storozhev
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.
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A non-Hopfian relatively hyperbolic group with respect to a Hopfian subgroup
本文通过构造一个相对于 Hopf 子群的非 Hopf 有限生成相对双曲群反例，否定回答了 Osin 提出的问题5.5。
A recipe for constructing non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups
本文利用像扩张定理提出了一种构造具有 Hopfian 边缘子群的 non-Hopfian 相对双曲群的一般方法，并给出了两个具体实例，为 Osin 问题提供了反例。
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。
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.
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。
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.
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。
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…
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。
We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such 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…
The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $ϕ$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $ϕ$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $ϕ$ app…
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。
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。
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。
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。
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.
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with …
Free-abelian by free groups: homomorphisms and algorithmic explorations
本文给出了自由阿贝尔乘自由群自同态的显式描述及其单射、满射判定，证明该类群是Hopfian且非coHopfian，并证明同构问题在有限作用下可判定、Brinkmann问题对单同态和自同构可判定。
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。
It is proved that an endomorphism $φ$ of a hyperbolic group $G$ satisfies a Hölder condition with respect to a visual metric if and only if $φ$ is virtually injective and $Gφ$ is a quasiconvex subgroup of $G$. If $G$ is virtually free or torsion-free co-hopfian, then $φ$ is uniformly continuous if and only if it it satisfies a Hölder condition if and only if it is virtually injective. Lipschitz conditions are discussed for free group automorphisms.
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。
In this paper we prove the theorem on freedom for relatively free groups with a single relation (analogous with the well-known result of Magnus) and the theorem on freedom for relatively free Lie algebras with a single relation (analogous with the well-known result of Shirshov).
The theorems on freedom for relatively free groups with a relations
本文证明了带单个关系的相对自由群的自由性定理，并推广了Freiheitssatz，分别类比马格努斯和罗曼诺夫斯基的著名结果。
The theorems on freedom for relatively free Lie algebras with a relations
本文针对相对自由李代数，证明了单关系情形的自由定理及广义Freiheitssatz，分别类比Shirshov与Kharlampovich的经典结果。
Open multiplication in relatively free profinite semigroupoids
本文将乘法为开等结果从相对自由射影有限半群推广至有限顶点图上的半群胚，并借此刻画无限字母表上的循环字及半群胚中的稳定子。
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…
The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.
We construct easy embeddings of relatively free groups (say the free Burnside group, the free solvable group) into finitely presented groups. We introduce a concept of verbal isoperimetric function of a group variety. We prove that if the verbal Dehn function of a relatively free group is bounded by a polynomial then the group can be embedded quasi-isometrically into a finitely presented group with polynomial isoperimetric function. We also construct an easy embedding of any Baumslag-Solitar solvable group into a finitely presented group with polynomial Dehn function.
On some relatively free pro-p groups
本文证明了在2个生成元上的相对自由中心-by-metabelian pro-p群与（2类幂零）-by-abelian pro-p群在所有profinite群类中均有限可公理化。
Self-similarity of some soluble relatively free groups
研究可解相对自由群的传递自相似性，证明有限秩自由幂零群具有该性质，而秩至少为2的自由亚Abel群不具有。
Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = {a_1,..., a_r} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {a_{l+1},...,a_r}, then S is a subset of a basis for the relatively free group on {a_1,...,a_l}.
We describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group.
We establish that, under certain closure assumptions on a pseudovariety of semigroups, the corresponding relatively free profinite semigroups freely generated by a non-singleton finite set act faithfully on their minimum ideals. As applications, we enlarge the scope of several previous join irreducibility results for pseudovarieties of semigroups, which turn out to be even join irreducible in the lattice of pseudovarieties of ordered semigroups, so that, in particular, they are not generated by proper subpseudovarieties of ordered semigroups. We also prove the stronger form of join irreducibil…
Free-by-cyclic groups are not relatively profinitely rigid
针对Bridson和Reid在2015年提出的问题，构造\(m = 5\)及\(m\geq7\)时两两非同构的循环自由群族，其同构于\(F_{2m}\rtimes\mathbb{Z}\)且有同构有限完备化。
We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela); and (ii) fully residually free groups (Bumagin, Kharlampovich and Miasnikov). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for $n \ge 3$. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabo…
Let F be an infinitely generated free group and R a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F' of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group Aut(F/R') of the group F/R' is complete. In particular, the automorphism group of any infinitely generated free solvalbe group of derived length at least two is complete. This extends a result by Dyer and Formanek (1977) on finitely generated groups F_n/R' where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.
Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a re…
Commutator relators of one-relator groups do not force Hopficity, residual finiteness, or automaticity
本文构造了一族单关系群，其关系为交换子，每个都有Baumslitar子群作为retract，从而证明了这些群不满足霍菲性、残余有限性和自动性。
We apply the method of Arzhantseva-Ol'shanskii to prove that for an exponentially generic (in the sense of Ol'shanskii) class of one-relator groups the isomorphism problem is solvable in at most exponential time. This is obtained as a corollary of our more general result that for any fixed integers $m>1, n>0$ there is an exponentially generic class of $m$-generator $n$-relator groups where every group has only one Nielsen equivalence class of $m$-tuples generating non-free subgroups. This means that a group $G$ in this class has has only one non-free $m$-generated subgroup, namely $G$ itself. …
We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $ψ$ and $ϕ$ with dual lamination pairs $Λ^\pm_ψ$ and $Λ^\pm_ϕ$ associated to them, and given a free factor system $\mathcal{F}$ with co-edge number $\geq 2$, $ϕ, ψ$ each preserving $\mathcal{F}$, so that the pair $(ϕ, Λ^\pm_ϕ), (ψ, Λ^\pm_ψ)$ is independent relative to $\mathcal{F}$, then there $\exists$ $M\geq 1$, such that for any integer $m,n \geq M$, the group $\langle ϕ^m, ψ^n \rangle$ is a free group of rank 2, all of whose non-trivial elements except perhaps the po…
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite subsystem, and a number of structural results about Gamma-limit groups.
Suppose $G$ is a 1-ended finitely presented group that is hyperbolic relative to $\mathcal P$ a finite collection of 1-ended finitely presented proper subgroups of $G$. Our main theorem states that if the boundary $\partial (G,{\mathcal P})$ is locally connected and the second cohomology group $H^2(P,\mathbb ZP)$ is free abelian for each $P\in \mathcal P$, then $H^2(G,\mathbb ZG)$ is free abelian. When $G$ is 1-ended it is conjectured that $\partial (G,\mathcal P)$ is always locally connected. Under mild conditions on $G$ and the members of $\mathcal P$ the 1-ended and local connectivity hypot…
Relative cohomological dimension of a relatively hyperbolic pair
该研究证明无挠相对双曲对的相对上同调维数有限，且在特定条件下拟等距不变，并通过所提方法计算了多类情形的该维数。
We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…
In this paper we describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon's completion $G^{\mathbb{Z}[t]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}[t]}$ containing $G$ is universally equivalent to $G$. Since finitely generated groups universally equivalent …
Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components
本研究将自由群与无挠双曲群的外自同构共轭问题归约为多项式增长自同构映射环面的相关算法问题，求解了多类新外自同构的共轭问题，为彻底解决$Out(F_n)$的共轭问题提供了可行路径。
Topological restrictions on relatively Anosov representations
本文利用 Bowditch 边界拓扑与 Anosov 极限映射的相互作用，给出群容许到 SL、Sp 等 Lie 群的相对 Anosov 表示时的群结构与维数限制。
Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.
A survey article that presents some recent algebraic and model-theoretic results on the automorphism groups of relatively free groups of infinite rank. The topics include topological aspects, generating sets, descripition of automorpisms and expressive power of the first-order theories.
We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $ϕ$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of subgroups in $\mathcal{G}$ on conjugates of themselves by the same element, gives rise to a semidirect product $G\rtimes_ϕ\mathbb{Z}$ that is relatively hyperbolic with respect to suspensions of groups in $\mathcal{G}$. We recover a theorem of Gautero-Lustig and Ghosh that, if $G$ is a free group, $ϕ$ an automorphism of $G$, and $\mathcal{G}$ is its family of polynomia…
We begin the investigation of Gamma-limit groups, where Gamma is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of Drutu and Sapir, we adapt the results from math.GR/0404440 to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate homomorphisms {h_n : G -> Gamma}, we extract an R-tree with a nontrivial isometric G-action. We then prove an analogue of Sela's shortening argument.
Let k be a local field, and G a linear group over k. We prove that either G contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups.
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures…
The profinite completion of relatively hyperbolic virtually special groups
本文利用profinite完备化刻画toral相对双曲的实际上特殊群，并证明其完备化子群的Tits二择一与有限生成pro-p子群结构，结果应用于双曲算术流形及同余核。
Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
本文证明了在QCERF相对双曲群中，有限生成相对拟凸子群存在有限指数子群使虚拟联接保持拟凸性，并由此导出多类群中子群乘积的可分性。
The aim of this note is to show that weak relative hyperbolicity of a group relative to a subgroup (or relative hyperbolicity in the sense of Farb) does not imply any natural analogues of some well-known algebraic properties of ordinary hyperbolic groups. Our main tools are combination theorems for weakly relatively hyperbolic groups.
In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.
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