Max Forester, Colin Rourke
In this note we prove injectivity and relative asphericity for "layered" systems of equations over torsion-free groups, when the exponent matrix is invertible over Z. We also give elementary geometric proofs of results due to Bogley-Pride and Serre that are used in the proof of the main theorem.
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We improve and shorten the argument given in(Journal of Algebra, vol.~210 (1998) pp~291--297). Inparticular, the fact that Artin braid groups are torsion free now follows from Garsideś results almost immediately.
We show that certain cyclically pinched one-relator groups are residually torsion-free nilpotent.
This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.
Let $X$ be a compact torsion abelian group. In this paper, we show that an extension of $F_{p}$ by $X$ splits where $F_{p}$ is the p-adic number group and $p$ a prime number. Also, we show that an extension of a torsion-free, non-divisible LCA group by $X$ is not split.
A finitely generated virtually free pro-p group with finite centralizers of its torsion elements is the free pro-p product of finite p-groups and a free pro-p factor.
We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotency of the free loop proved by Higman, it is proved, essentially, by the same method.
Types in torsion free Abelian groups
该研究聚焦无扭阿贝尔群的逻辑型与等型等价问题，刻画了群内元素及标准二元组的所有可能型，完成了可分无扭阿贝尔群的等型分类。
A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch length at most 5.
A second countable virtually free pro-p group all of whose torsion elements have finite centralizer is the free pro-p product of finite p-groups and a free pro-p factor.
We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.
On singular equations over torsion-free groups
本文证明无挠群单关系子积的Freiheitssatz（音节长度≤8），并应用于无挠群上音节长度≤18的奇异方程，证明其在某个超群中有解。
We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free
We construct examples of groups showing that virtual solvability and the property of being virtually torsion-free are not preserved by bi-Lipschitz maps and hence by quasi-isometries.
Let H and K be quasiconvex subgroups of a negatively curved torsion-free group G. We give an algorithm which decides whether an element of H is conjugated in G to an element of K.
A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion free nilpotent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.
Some torsion-free solvable groups with few subquotients
本研究构造出满足所有有限生成无挠亚阿贝尔子商均为虚阿贝尔的无限秩有限生成无挠可解群，证明了P. Kropholler的无限秩可解群刻画定理不存在无挠版本。
We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.
We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.
We use a recent result of Alexander and Nishinaka to show that if $G$ is a non-elementary torsion-free hyperbolic group and $R$ is a countable domain, then the group ring $RG$ is primitive. This implies that the group ring $KG$ of any non-elementary torsion-free hyperbolic group $G$ over a field $K$ is primitive.
Generalized torsion in amalgams
本文给出融合积为广义无挠的充分条件，并据此构造出闭3-流形基本群、单关系群及一般群等广义无挠但不可序的例子。
We introduce a condition on the monodromy of a free-by-cyclic group, $G_ϕ$, called the chain flare condition, that implies that the $L^2$-torsion, $ρ^{(2)}(G_ϕ)$, is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.
It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.
In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$
An indecomposable decomposition of a torsion-free abelian group $G$ of rank $n$ is a decomposition $G=A_1\oplus\cdots\oplus A_t$ where $A_i$ is indecomposable of rank $r_i$ so that $\sum_i r_i=n$ is a partition of $n$. The group $G$ may have decompositions that result in different partitions of $n$. We address the problem of characterising those sets of partitions of $n$ which can arise from indecomposable decompositions of a torsion-free abelian group.
Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.
We give a complete characterization of torsion-free hyperbolic groups which are homogeneous in the sense of first-order logic, in terms of the JSJ decompositions of their free factors.
In the presented paper we consider some methods of studying of prime ideals in group algebras of abelian groups of finite rank
Uncountably many quasi-isometric torsion-free groups
本文借助Schur覆盖群、群上同调及有界值上同调理论，构造出不可数个同属一个拟等距类、有限生成且两两不同构的无挠群。
In this paper we study the property of geometric equivalence of groups introduced by B. Plotkin \cite{P1, P2}. Sufficient and necessary conditions are presented for a $π$-torsion-free nilpotent group to be geometrically equivalent to its $π$-completion. We prove that a relatively free nilpotent $π$-torsion-free group and its $π$-completion define the same quasi-variety. Examples of $π$-torsion-free nilpotent groups that are geometrically equivalent to their $π$-completions are given.
A proof of the Borel completeness of torsion free abelian groups is presented. This proof differs considerably from the approach of Paolini-Shelah.
We fill details in the proof of Lemma 13 in Bull.London.Math.Soc 40 (2008) (that is Lemma 3.2 in arXiv:0712.4244v1).
Let G be a torsion free hyperbolic group. We prove that the elementary theory of G is decidable and admits an effective quantifier elimination to boolean combination of AE-formulas. The existence of such quantifier elimination was previously proved by Sela.
In this note, we construct torsion-free countable, amenable, weakly mixing groups, which answer a question of V. Bergelson. Some results related to verbal subgroups and crystallographic groups are also presented.
On self-similarity of finitely generated torsion-free nilpotent groups
本文研究有限生成无挠幂零群的自相似性，建立了几何等价性与自相似性的联系，证明所有2生成3类无挠幂零群可忠实表示为传递自相似群，并给出一个4生成3类反例，解决了Sidki提出的问题。
A torsion-free non-sofic group
该研究在 OpenAI 发现二元 Leavitt 代数的单位群为非 Sofic 群的基础上，提供了另一类含无挠群的非 Sofic 群例子，依托相同技术准则。
Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly indecomposable groups as a subgroup of finite index.
We find a necessary condition for zero divisors in complex group algebras of torsion-free groups.
Let $Γ$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $Γ$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $Γ$ is free.
$R_{\infty}$-property for finitely generated torsion-free 2-step nilpotent groups of small Hirsch length
本文研究了有限生成无挠2步幂零群的$R_{\infty}$-性质，证明Hirsch长度至多为6的群不具有该性质，而长度为7的群存在具有该性质的例子。
We generalize the graphical small cancellation theory of Gromov to a graphical small cancellation theory over the free product. We extend Gromov's small cancellation theorem to the free product. We explain and generalize Rips-Segev's construction of torsion-free groups without the unique product property by viewing these groups as given by graphical small cancellation presentations over the free product. Our graphical small cancellation theorem then provides first examples of Gromov hyperbolic groups without the unique product property. We construct uncountably many non-isomorphic torsion-free…
In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more comprehend the complication of the problem of the classification of the quasi-varieties of nilpotent class 2 groups. It is well known that the variety of a nilpotent class s (for every s) groups is Noethe…
Uniform small cancellation in torsion-free hyperbolic groups
该研究针对非初等无扭双曲群建立一致小消去定理，并将其应用于证明这类群的一阶理论不具有模型伴侣。
A nontrivial commutator cannot be a proper power in a torsion-free group satisfying C'(λ) small cancellation condition with sufficiently small λ.
We give a complete survey of a construction by Boone and Collins for embedding any finitely presented group into one with $8$ generators and $26$ relations. We show that this embedding preserves the set of orders of torsion elements, and in particular torsion-freeness. We combine this with the independent results of Belegradek and Chiodo to prove that there is an $8$-generator $26$-relator universal finitely presented torsion-free group (one into which all finitely presented torsion-free groups embed).
Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.
We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and …
We study some close relationships between the classes of transitive, fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some oldstanding problems, posed by Krylov-Mikhalev-Tuganbaev in their monograph [17]. Specifically, we answer Problem 44 from there in the affirmative by constructing a Krylov transitive torsion-free Abelian group which is neither fully transitive nor transitive. This extends to the torsion-free case certain similar results in the p-torsion case, obtained by Braun et al. in J. Algebra (201…
We describe the structure of ``K-approximate subgroups'' of torsion-free nilpotent groups, paying particular attention to Lie groups. Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which independently address related issues. We comment briefly on some of the connections between these papers.
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…
We prove that the generic type of a non-cyclic torsion-free hyperbolic group G is foreign to any interpretable abelian group, hence also to any interpretable field. This result depends, among other things, on the definable simplicity of a non-cyclic torsion-free hyperbolic group, and we take the opportunity to give a proof of the latter using Sela's description of imaginaries in torsion-free hyperbolic groups. We also use the description of imaginaries to prove that if F is a free group of rank > 2 then no orbit of a finite tuple from F under Aut(F) is definable.
We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups elementarily embedded in finitely generated free groups are free factors.
The OP subgroup of a torsion-free LCA group
本文定义了局部紧阿贝尔（LCA）群的$OP$子群，并证明无挠LCA群的$OP$子群是一个非零纯子群。
Torsion homology growth of polynomially growing free-by-cyclic groups
本研究证明多项式增长单值群的自由循环群同调挠增长各维度均为零且与Farber链选取无关，推导出整挠等于ℓ²-挠，验证了Lück的相关猜想。
A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topolo…
A torsion-free group of nilpotency class six with all subgroups subnormal of defect at most $5$
该研究构造了一个2生成的无扭幂零群，其幂零类为6且所有子群均为亏格至多5的次正规，从而否定了Casolo关于此类群幂零类与亏格关系的问题在n=5时的结论。
We study the torsion free generalized crystallographic groups with the indecomposable holonomy group which is isomorphic to either a cyclic group of order ${p^s}$ or a direct product of two cyclic groups of order ${p}$.
Zero divisors and units with small supports in group algebras of torsion-free groups
本文提出一种图论方法，将无挠群群代数中可能的非零零因子与图相关联进行研究。
We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $Γ$, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from $G$ to $Γ$ as compositions of factorizations through $Γ$-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of $Γ$-limit groups as iterated generalized doubles over $Γ$.
Torsion in asymptotic cones of free groups
该研究证明了配备双不变字度量的任意非阿贝尔自由群的渐近锥存在2阶元素，丰富了群的渐近几何性质研究。
This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.
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