D. Tieudjo, D. I. Moldavanskii
The description of the automorphism group of group $<a, b; [a^m,b^n]=1>$ ($m,n>1$) in terms of generators and defining relations is given. This result is applied to prove that any normal automorphism of every such group is inner.
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The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.
It is a well-known fact that every group $G$ has a presentation of the form $G = F/R$, where $F$ is a free group and $R$ the kernel of the natural epimorphism from $F$ onto $G$. Driven by the desire to obtain a similar presentation of the group of automorphisms $Aut(G)$, we can consider the subgroup $Stab(R) \subseteq Aut(F)$ of those automorphisms of $F$ that stabilize $R$, and try to figure out if the natural homomorphism $Stab(R) \to Aut(G)$ is onto, and if it is, to determine its kernel. Both parts of this task are usually quite hard. The former part received considerable attention in the …
We prove the following results: (1) There is a one-relator inverse monoid $\mathrm{Inv}\langle A\:|\:w=1 \rangle$ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The first of these results answers a problem originally posed by Margolis, Meakin and Stephen in 1987.
We show that certain cyclically pinched one-relator groups are residually torsion-free nilpotent.
This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sification efforts we single out some relator types that render the corresponding one-relator groups residually solvable.
Hyperbolic one-relator groups
本文提出原始扩展群的概念，通过刻画超几何原始扩展群来解决超几何一关系群的分类问题，并利用Christoffel词和一关系塔机制进行证明。
A one-relator group with lower central series of length ω^2 is constructed. This answers a problem of G. Baumslag.
The word problem for two-generator one-relator inverse monoids
本文研究两生成元单关系逆幺半群的字问题，证明了其可判定与不可判定结果，给出首个不可判定字问题的E-酉两生成元单关系逆幺半群等例子，还给出其可判定字问题的充分条件。
We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov and give an estimate of their asymptotic dimension in terms of the relator length.
More Residually Solvable One-Relator Groups
证明具有部分正关系词的一关系群是剩余可解的，推广了Baumslag关于正关系词情形的定理。
Conjugacy separability of any group of the class of one-relator groups given by the presentation $<a, b; [a^m,b^n]=1>$ ($m,n>1$) is proven.
A one-relator surface group is the quotient of an orientable surface group by the normal closure of a single relator. A Magnus subgroup is the fundamental group of a suitable incompressible sub-surface. A number of results are proved about the intersections of such subgroups and their conjugates, analogous to results of Bagherzadeh, Brodskii, and Collins in classical one-relator group theory.
In this paper we prove several results regarding decidability of the membership problem for certain submonoids in amalgamated free products and HNN extensions of groups. These general results are then applied to solve the prefix membership problem for a number of classes of one-relator groups which are low in the Magnus-Moldavanski\uı hierarchy. Since the prefix membership problem for one-relator groups is intimately related to the word problem for one-relator special inverse monoids in the $E$-unitary case (as discovered in 2001 by Ivanov, Margolis and Meakin), these results yield solutions o…
We prove that one-relator groups with torsion are hereditarily conjugacy separable. Our argument is based on a combination of recent results of Dani Wise and the first author. As a corollary we obtain that any quasiconvex subgroup of a one-relator group with torsion is also conjugacy separable.
Residually rationally solvable one-relator groups
本文证明单关系群的有理导出列之交有理完美且由单个元素正规生成，并由此精确刻画了单关系群剩余有理可解的条件。
In this paper, we generalise Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.
The general {\bf surface group conjecture} asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in [FKMRR]. First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if $G$ is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then $G$ is free, a surface group or a solvable Baumslag-Solitar Group. Further combining results …
In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of one-relator groups with torsion closely resemble the structure of torsion-free hyperbolic groups.
The focus of this article is to define the descriptively approximations in proximal relator spaces. Afterwards, descriptive approximately algebraic structures such as groupoids, semigroups and groups in digital images endowed with descriptive Efremovic proximity relations were introduced.
Rank 1 phenomena for 1-relator and deficiency 1 groups
本文通过研究有限指数子群的有界上同调和模p同调，精确刻画了1-关系群及亏量至少1的群的有界生成性，并排除了猜想中不存在的例外类。
We consider largeness of groups given by a presentation of deficiency 1, where the group is respectively free-by-cyclic, LERF or 1-relator. We give the first examples of (finitely generated free)-by-(infinite cyclic) word hyperbolic groups which are large, show that a LERF deficiency 1 group with first Betti number at least 2 is large or the integers times the integers, and show that 2-generator 1-relator groups where the relator has height 1 obey the dichotomy that either the group is large or all its finite images are metacyclic.
A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.
Quasi-convex surface subgroups in some one-relator groups with torsion
本文在特定带挠的一关系群中构造了面子群，并利用这一结果推导了自由群中一个词为原始词的一个profinite准则。
The stable commutator length of a relator is not a one-relator group invariant
该研究通过构造两个同构的单关系群，证明关系子的稳定换位子长度并非单关系群不变量，否定了Heuer与Löh提出的相关问题。
Margulis-Soifer theorem for one-relator groups
本文证明了单关系群满足Margulis-Soifer二分法，即每个单关系群要么几乎可解，要么有无限指数的极大子群，并给出相关例子及Frattini子群的平凡性结果。
A new family of sofic one-relator groups
本文构造了一族无限的sofic单关系群，并证明它们既非剩余可解也非剩余有限，其证明不依赖Magnus分解，与已有方法本质不同。
Short, highly imprimitive words yield hyperbolic one-relator groups
本文为 Louder 和 Wilton 的猜想提供了实验支持，验证了非本原秩大于二的字能生成双曲单关系群。
A note on right-angled Artin subgroups of one-relator groups
本文通过基本的Bass-Serre理论和一个关系群的经典性质，证明了当右角艺术群嵌入到一个一个关系群中时，Γ是一个有限森林。
We prove a rank 1 version of the Hanna Neumann Theorem. This shows that every one-relator 2-complex without torsion has the nonpositive immersion property. The proof generalizes to staggered and reducible 2-complexes.
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。
In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus method and Composition-Diamond Lemma, we reprove the G. Higman, B. H. Neumann and H. Neumann's embedding theorem.
On groups of units of special and one-relator inverse monoids
该研究探究特殊与单关系逆幺半群的单位群，提出求单位群生成表的新方法并给出其为单关系群的充分条件，同时构造反例证明相关经典结果对逆幺半群一般不成立。
A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the re…
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 …
We continue research into the cyclically presented groups with length three positive relators. We study small cancellation conditions and SQ-universality, we obtain the Betti numbers of the groups' abelianisations, we calculate the orders of the abelianisations of some groups, and we study isomorphism classes of the groups. Through computational experiments we assess how effective the abelianisation is as an invariant for distinguishing non-isomorphic groups.
Gromov's Conjecture for Product of Baumslag-Solitar groups and some other One-relator groups
证明Baumslag-Solitar群及部分推广不可定向曲面群的单关系群是虚拟2-可避免的，结合前期结果，证实了基本群为这些群乘积的闭定向自旋流形的Gromov PSC流形万有覆盖宏观维数猜想。
On pro-${\it p}$-groups with a single defining relator
该研究针对群论领域的pro-p群相关问题，给出了Magnus自由定理可推广到自由pro-p群的必要条件，推进了相关群结构理论的发展。
A class of one-relator groups such that every group in the class is determined by a triple of integers and is an HNN-extension of some Baumslag -- Solitar group is considered. A criterion for two groups in this class to be isomorphic and criterion for two non-isomorphic groups to be a homomorphic image of each other are obtained. This result leads to a negative answer to the question 3.33 from "Kourovka Notebook".
We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also point out a variation of the construction that yields examples of lattices in $\tilde A_2$-buildings admitting non-Desarguesian residues of arbitrary prime power order.
Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance, adding one generator and one relator in which the exponent sum of the additional generator is one to a free product of two nontrivial torsion-free groups, we also obtain an SQ-universal group. Key words: relative presentations, one-relator groups, SQ-universality, equations over groups.
Long relators in groups generated by two parabolic elements
本文研究两个抛物元素生成的群的长关系子特性，提出两种适用于非自由群的乒乓引理变体，证明存在关系子音节数任意大的群，指出现有相关方法的局限性。
Exponential equations in free groups were studied initially by Lyndon and Schutzenberger and then by Comerford and Edmunds. Comerford and Edmunds showed that the problem of determining whether or not the class of quadratic exponential equations have solution is decidable, in finitely generated free groups. In this paper we show that for finite systems of quadratic exponential equations decidability passes, under certain hypotheses, from the factor groups to free products and one-relator products.
The main achievement of this thesis is an algorithm which given a finite group presentation and natural numbers n and k, computes all the relators of length and area up to n and k respectively. The complexity of this algorithm is better by a factor which is over-exponential than that of classical methods using van Kampen diagrams.
All hyperbolic cyclically presented groups with positive length three relators
本文完整分类了由正长度三关系子定义的双曲循环呈现群，证明其双曲性当且仅当参数 m 属于 {1,2,3,6,9}。
Asymptotic Dimension of Graphs of Groups and One Relator Groups
本文证明HNN-扩张渐近维数的新不等式，证实单关系子群渐近维数至多为2的猜想，并计算右角Artin群精确维数及给出图的基本群的新上界。
Undecidable Diophantine problems in generalisations of one-relator groups
本文通过构造具体例子，证明了若干与单关系子群接近的群族具有不可判定的丢番图问题，包括广义Baumslag-Solitar群、单关系循环群乘积、带约束的单关系群以及自由-自由群。
Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.
Correction: Root extraction in one-relator groups and slenderness
本更正论文指出早前一篇单关系群根提取与细性相关论文的论证疏漏，并给出一个与原论文主结果风格相近的定理。
We prove that Whitehead's algorithm for solving the automorphism problem in a fixed free group $F_k$ has strongly linear time generic-case complexity. This is done by showing that the ``hard'' part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type isomorphism rigidity result for random one-relator groups: If two such groups are isomorphic then their Cayley graphs on the \emph{given generating sets} are isometric. Although no nontrivial examples were previously known, we prove tha…
It is shown that the strong Atiyah conjecture and the Lück approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.
Let H be a group obtained from a group G by adding one generator and one relator which is a proper power of a unimodular word. The main aim of this paper is to prove that G is a malnormal subgroup of H. This implies that if G is a nontrivial group, then the group H has a trivial center.
For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 α}$ where $α= \log_2(2p / q)$. The group $R_{p,q}$ has no subgroup isomorphic to a Baumslag-Solitar group $BS(m,n)$ with $m \neq \pm n$, but is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.
An embedding construction $G\hookrightarrow H$ for groups $G$ with a length function was introduced by the author earlier. Here we obtain new properties of this embedding, answering some questions raised by M.V. Sapir. In particular, an analog of Tits' alternative holds for the subgroups of $H$.
G.E. Wall gave two different proofs of a remarkable result about the multilinear Lie relators satisfied by groups of prime power exponent $q$. He showed that if $q$ is a power of the prime $p$, and if $f$ is a multilinear Lie relator in $n$ variables where $n\neq1\operatorname{mod}(p-1)$, then $f=0$ is a consequence of multilinear Lie relators in fewer than $n$ variables. For years I have struggled to understand his proofs, and while I still have not the slightest clue about his first proof published in the Journal of Algebra, I finally have some understanding of his second proof published in …
Soficity of One-Relator Groups and Reducible Presentations
本文证明了所有有限生成的单关系群均为 sofic 群，通过终端路径上的自由群单词构造上循环，并利用斜积作用实现有限置换逼近。
In the theory of one-relator groups, Magnus subgroups, which are free subgroups obtained by omitting a generator that occurs in the given relator, play an essential structural role. In a previous article, the author proved that if two distinct Magnus subgroups M and N of a one-relator group, with free bases S and T are given, then the intersection of M and N is either the free subgroup P generated by the intersection of S and T or the free product of P with an infinite cyclic group. The main result of this article is that if M and N are Magnus subgroups (not necessarily distinct) of a one-rela…
On the coherence of one-relator groups and their group algebras
本文证明了单关系群的凝聚性，解决了 Gilbert Baumslag 的著名问题，并将该策略应用于上同调维数为2的群类，证明了二维复形基本群、交错展示群、部分 Coxeter 群及其群代数的凝聚性。
Non-triviality of some one-relator products of three groups
本文研究三个非平凡群自由积关于单元素正规闭包的商群G，证明若关系元长度至多为8则G非平凡，并在循环群因子情形下证明至少一个因子嵌入G。
We introduce the notion of strictly systolic angled complexes. They generalize Januszkiewickz and Świątkowski's $7$-systolic simplicial complexes and also their metric counterparts, which appear as natural analogues to Huang and Osajda's metrically systolic simplicial complexes in the context of negative curvature. We prove that strictly systolic angled complexes and the groups that act on them geometrically, together with their finitely presented subgroups, are hyperbolic. We use these complexes to study the geometry of one-relator groups without torsion, and prove hyperbolicity of such group…
For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying presentations. We show that the fixed point subgroup of the shift $x_i \mapsto x_{i+1}$ is always finite and we relate finiteness of $G$ and asphericity to the dynamics of the shift action by the cyclic group of order $n$ on the nonidentity elements of $G$.
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