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For an ascending HNN-extension $G*_{\psi}$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_{\psi}}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-colorings, $n\ge 3$, of the (right) Cayley graph of $\mathrm{BS}(1,N)$, estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm{BS}(1,N)$ admits a frozen $n$-coloring if and only if $n=3$. We finally suggest generalizations of the latter results to $n$-colorings of ascending HNN-extensions of finitely generated abelian groups., Comment: 22 pages

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$, where $C$ is a finite group and $F$ a non-abelian free group.

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb H^4$ with fundamental domain the $120$-cell or the $24$-cell., 30 pages, to appear in J. Inst. Math. Jussieu

We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated: We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated, and on the other hand construct a finitely generated, infinite dimensional nil algebra whose adjoint group is generated by elements of bounded torsion., To appear in Proceedings of the Edinburgh Mathematical Society

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincar\'e series. As an application, we show that the Poincar\'e series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator., Comment: 22 pages; new Section 7 incorporating DGAlgebras Macaulay2 package, updated Theorems 3.1 and 6.1 to improve and clarify generation property, and some more minor adjustments

Motivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

We associate with every etale groupoid G two normal subgroups S(G) and A(G) of the topological full group of G, which are analogs of the symmetric and alternating groups. We prove that if G is a minimal groupoid of germs (e.g., of a group action), then A(G) is simple and is contained in every non-trivial normal subgroup of the full group. We show that if G is expansive (e.g., is the groupoid of germs of an expansive action of a group), then A(G) is finitely generated. We also show that S(G)/A(G) is a quotient of H_0(G, Z/2Z)., 25 pages, 3 figures

Some classes of finitely generated hyperabelian groups defined in terms of semipermutability and S-semipermutability are studied in the paper. The classification of finitely generated hyperabelian groups all of whose finite quotients are PST-groups recently obtained by Robinson is behind our results. An alternative proof of such a classification is also included in the paper.

Over a Cohen–Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a “structure theorem” for balanced big CM modules. (2) Every finitely generated module has a pre-envelope with respect to the class of finitely generated maximal CM modules. This result is, in some sense, dual to the existence of maximal CM approximations, which has been proved by Auslander and Buchweitz.

We prove that an operator system is (min, ess)-nuclear if its C*-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of countable discrete group by Farenick et al. is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (resp., a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to 3 (resp., every component of the graph is complete)., Comment: To appear in J. Aust. Math. Soc

Let $A$ be a local complete intersection ring. Let $M,N$ be two finitely generated $A$-modules and $I$ an ideal of $A$. We prove that \[ \bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}\mathrm{Ass}_A\left(\mathrm{Ext}_A^i(M,N/I^n N)\right) \] is a finite set. Moreover, we prove that there exist $i_0,n_0\geqslant 0$ such that for all $i\geqslant i_0$ and $n \geqslant n_0$, we have \[ \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \] \[ \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i+1}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \] We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity $\mathrm{cx}_A(M,N/I^nN)$ is constant for all sufficiently large $n$., Comment: 17 pages, final version

Let$G$be a finitely generated amenable group. We study the space of shifts on$G$over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that, more generally, the shifts of entropy$c$are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that, for every entropy value$c\in [0,\log |A|]$, there is a weakly mixing subshift of$A^{G}$with entropy $c$. We also show that the set of strongly irreducible shifts does not form a$G_{\unicode[STIX]{x1D6FF}}$in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

We prove that a non-solvable Demushkin group satisfies the Greenberg–Stallings property, i.e., if H and K are finitely generated subgroups of a non-solvable Demushkin group G with the property that H ∩ K has finite index in both H and K, then H ∩ K has finite index in 〈H, K〉. Moreover, we prove that every finitely generated subgroup H of G has a ‘root’, that is a subgroup K of G that contains H with |K : H| finite and which contains every subgroup U of G that contains H with |U : H| finite. This allows us to show that every non-trivial finitely generated subgroup of a non-solvable Demushkin group has finite index in its commensurator.

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

We study when a pro-p subdirect product S ⩽ G1 × . . . × Gn is of type FPm for m ⩾ 2 for some special pro-p groups Gi. In particular we treat the case when Gi is a finitely generated non-trivial free pro-p product different from C2 ∐ C2 if p = 2 or a non-abelian pro-p group from the class $\mathcal{L}$ defined in [12].

Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

We show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.

We give a complete characterization of a hypercyclic abelian semigroup of matrices on ℂn. For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over ℂ that form a hypercyclic abelian semigroup on ℂn. In particular, we show that no abelian semigroup generated by n matrices on ℂn can be hypercyclic.

Let $\mathcal{C}$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if u : H ↪ G is a map of finitely generated residually $\mathcal{C}$ groups such that the induced map û : Ĥ → Ĝ is a surjection of the pro-$\mathcal{C}$ completions, and G has more than one end, then H has the same number of ends as G. However if G has one end the number of ends of H may be larger; we observe cases where this occurs for $\mathcal{C}$ the class of finite p-groups.We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or û : Ĥ → Ĝ is an isomorphism of profinite completions but H has property (T) (and hence (FA)), but G has neither. Either possibility would give new examples of pathological finitely generated groups.

In this paper we describe the well-founded initial segment of the free Heyting algebra α on finitely many, α, generators. We give a complete classification of initial sublattices of 2 isomorphic to 1 (called ‘low ladders’), and prove that for 2 ≤ α < ω, the height of the well-founded initial segment of α is ω2.

We show that, for finitely generated groups, the property of admitting a coarse median structure is preserved under relative hyperbolicity.

Let G be a group. We say that G∈𝒯(∞) provided that every infinite set of elements of G contains three distinct elements x,y,z such that x≠y,[x,y,z]=1=[y,z,x]=[z,x,y]. We use this to show that for a finitely generated soluble group G, G/Z2(G) is finite if and only if G∈𝒯(∞).

Let f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight k and a unitary multiplier system v on a subgroup Γ ⊂ SL2(ℝ) that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n ≥ 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group Γ0*(N) (N ∈ ℕ) generated by the Hecke congruence subgroup Γ0(N) and the Fricke involution $W_N:= \big(\scriptsize\begin{array}{c@{}c} 0 & -{1/\sqrt N} \\[3pt] \sqrt N & 0\\ \end{array}\big)$ provided that the associated period functions are polynomials.

Let R be a commutative ring. It is shown that if an R-module M is a sum of δ-local submodules and a semisimple projective submodule, then every finitely generated submodule of M is δ-supplemented. From this result, we conclude that finitely generated δ-supplemented modules over commutative rings are amply δ-supplemented.

Let {Gr,i} be a sequence of r-generator Kleinian groups acting on ${\overline {\mathbb {R}}}^n$. In this paper, we prove that if {Gr,i} satisfies the F-condition, then its algebraic limit group Gr is also a Kleinian group. The existence of a homomorphism from Gr to Gr,i is also proved. These are generalisations of all known corresponding results.

We provide a general procedure for computing the algebraic K-theory of finitely generated virtually free groups. The procedure describes these groups in terms of the algebraic K-theory of various finite subgroups and various Farrell Nil groups. We illustrate this process by carrying out the computation for several interesting classes of examples. The first two classes serve as a check on the method and show that our algorithm recovers results that already exist in the literature. The last two classes of examples yield new computations.

Generalizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

Let R = ⊕i ∈ ℤRi be a ℤ-graded ring and M = ⊕i ∈ ℤMi be a graded R-module. Providing some results on graded multiplication modules, some equivalent conditions for which a finitely generated graded R-module to be graded multiplication will be given. We define generalised graded multiplication module and determine some of its certain graded prime submodules. It will be shown that any graded submodule of a finitely generated generalised graded multiplication R-module M has a kind of primary decomposition. Using this, we give a characterisation of graded primary submodules of M. These lead to a kind of characterisation of finitely generated generalised graded multiplication modules.

A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

For each k ≥ 2, we determine the asymptotic behaviour of the sequence of cardinalities of finitely generated free objects in , the variety consisting of all k-testable semigroups.

Let R be a commutative ring with identity. We will say that an R-module M has Nakayama property, if IM = M, where I is an ideal of R, implies that there exists a ∈ R such that aM = 0 and a − 1 ∈ I. Nakayama's Lemma is a well-known result, which states that every finitely generated R-module has Nakayama property. In this paper, we will study Nakayama property for modules. It is proved that R is a perfect ring if and only if every R-module has Nakayama property (Theorem 4.9).

We call a module Malmost perfect if every M-generated flat module is M-projective. Any perfect module is almost perfect. We characterize almost-perfect modules and investigate some of their properties. It is proved that a ring R is a left almost-perfect ring if and only if every finitely generated left R-module is almost perfect. R is left perfect if and only if every (projective) left R-module is almost perfect.

A variety is said to be a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees–Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees–Sushkevich varieties, the set ℱ of finitely generated varieties constitutes an incomplete sublattice and the set 𝒮 of small varieties constitutes a strict incomplete sublattice of ℱ. Consequently, a combinatorial Rees–Sushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set Σ of identities defines, within the largest combinatorial Rees–Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity 𝒪(nk) where n is the number of identities in Σ and k is the length of the longest word in Σ.

J. W. Anderson (1996) asked whether two finitely generated Kleinian groups $G_{1}, G_{2}\subset \mathrm {Isom}(\mathbb {H}^{n})$ with the same set of axes are commensurable. We give some partial solutions.

An element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

Let M be a module over the ring R. The finitary automorphism group of M over R is and the Artinian-finitary automorphism group of M over R is We continue our study begun in [9] and [12] of the relationship between these two in some ways similar, but actually quite different types of automorphism group. Our study here involves the generalized finitary group $\{{\rm g} \in {\rm Aut}_{\rm R}{\rm M}{:}\,{\rm M}({\rm g} - 1)$ embeds into some finitely generated R-module which does not fit our earlier pattern of generalized finitary groups as described in [8].

We construct the finitely generated free algebras and determine the free spectra of varieties of linear equivalential algebras and linear equivalential algebras of finite height corresponding, respectively, to the equivalential fragments of intermediate Gödel-Dummett logic and intermediate finite-valued logics of Gödel. Thus we compute the number of purely equivalential propositional formulas in these logics in n variables for an arbitrary n ∈ ℕ.

We determine those pseudovarieties of groups ${\bf H}$ for which the power monoids $P(G)$ , ranging over all groups $G$ in ${\bf H}$ , satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of ${\cal J}$ -trivial monoids by groups in ${\bf H}$ . That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation ${\bf PH}\,{=}\,{\bf J}\ast {\bf H}$ . The characterization is in terms of the geometry of the Cayley graphs of the free pro- ${\bf H}$ groups as well as in terms of the pro- ${\bf H}$ topology of a finitely generated free group.

1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type p′ interbounded with p, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.2. If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σω, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while fixing Σ generically In case p is Σ-internal and T is stable, this is the binding group of p over Σ.

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

This self-contained paper is part of a series (Groups of homeomorphisms of one-manifolds I: actions of nonlinear groups. Preprint , 2001; Group actions on one-manifolds II: extensions of Holder's Theorem. To appear in Trans. Amer. Math. Soc. ) seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante and Thurston proved that every nilpotent subgroup of Diff 2 ( S 1 ) is abelian. One of our main results is a sharp converse: Diff 1 ( S 1 ) contains every finitely generated, torsion-free nilpotent group.

We prove that any finitely generated solvable group of zero entropy contains a nilpotent subgroup of finite index. In particular, any finitely generated solvable group of exponential growth is of uniformly exponential growth.

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

LetNbe a closeds-Hopfiann-manifold with residually finite, torsion free π1(N) and finiteH1(N). Suppose that either πk(N) is finitely generated for allk≥ 2, or πk(N) ≅ 0 for 1

A group $G$ is called a group with boundedly finite conjugacy classes (or a BFC-group) if $G$ is finite-by-abelian. A group $G$ satisfies the maximal condition on non-BFC-subgroups if every ascending chain of non-BFC-subgroups terminates in finitely many steps. In this paper the authors obtain the structure of finitely generated soluble-by-finite groups with the maximal condition on non-BFC subgroups.AMS 2000 Mathematics subject classification: Primary 20E15. Secondary 20F16; 20F24

The paper shows that specializations of finitely generated graded modules are also graded and that many important invariants of graded modules and ideals are preserved by specializations.AMS 2000 Mathematics subject classification: Primary 13A02