Testing polycyclicity of finitely generated rational matrix groups

We describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of and thus we show that these properties are decidable. Variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of .

     

Representation dimension and finitely generated cohomology

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.     

On finitely generated idempotent semigroups

We give two short proofs of the well-known fact that every finitely generated idempotent semigroup is finite.     

Finitely generated permutative varieties

We show that there exists an algorithm which decides whether a finite set of identities containing a permutation identity defines a variety generated by a finite semigroup or not.     

Finitely additive random walks on infinitely generated free groups

We consider any purely finitely additive probability measure supported on the generators of an infinitely generated free group and the Markov strategy with stationary transition probability . As well as for the case of random walks (with countably additive transition probability) on finitely generated free groups, we prove that all bounded sets are transient. Finally, we consider any finitely additive measure (supported on the group generators) and we prove that the classification of the state space depends only on the continuous part of .     

Uncertainty principles in finitely generated shift-invariant spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in L²(R^d)$L^2({\mathbb{R}}^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in L²(R)$L^2(\mathbb{R})$ whose generator and its translates form a Riesz basis.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.     

Finitely generated submodels of an uncountably categorical homogeneous structure

We generalize the result of non‐finite axiomatizability of totally categorical first‐order theories from elementary model theory to homogeneous model theory. In particular, we lift the theory of envelopes to homogeneous model theory and develope theory of imaginaries in the case of ω‐stable homogeneous classes of finite U‐rank. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finitely generated subgroups of free profinite groups and of some Galois groups

It is proved that, for a (closed) subgroupH of a free profinite or free prosolvable groupF of rankF>1 such thatH contains a nontrivial composition subgroupN ofF, we have rankF< and [F:H]<.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 95–106, July, 1998.     

On monomial representations of finitely generated nilpotent groups

A result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.     

Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

Let G be a group and Aut_c(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Aut_c(G) = Inn(G) if and only if Z(G) = G′ and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G′ =Z(G) is infinite cyclic and Inn(G) < C* = Aut_c(G). In this article, we characterize all finitely generated groups G for which the equality Aut_c(G) = Inn(G) holds.     

Actions of finitely generated groups on the universal Menger curve

We prove that every finitely generated group acts effectively on the universal Menger curve.

     

Metric spaces are universal for bi-interpretation with metric structures

In the context of metric structures introduced by Ben Yaacov, Berenstein, Henson, and Usvyatsov [3], we exhibit an explicit encoding of metric structures in countable signatures as pure metric spaces in the empty signature, showing that such structures are universal for bi-interpretation among metric structures with positive diameter. This is analogous to the classical encoding of arbitrary discrete structures in finite signatures as graphs, but is stronger in certain ways and weaker in others. There are also certain fine grained topological concerns with no analog in the discrete setting.     

An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety [𝔑 _{c 1} , 𝔑 _{c 2} ], for all c ₂ ≤ c ₁ ≤ 2c ₂. As a consequence, we determine necessary and sufficient conditions for such groups to be [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable. We also show that if c ₁ ≠ 1 ≠ c ₂, then a finitely generated abelian group is [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable if and only if it is capable. Finally, we show that 𝔖₂-capability implies capability, but there is a capable finitely generated abelian group which is not 𝔖₂-capable.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

无限循环群被有限生成Abel群的中心扩张

设G是无限循环群被有限生成Abel群的中心扩张,T是G的中心ζG的挠子群.如果T的阶与ζG/(G'⊕T)的挠子群的阶互素,那么群G可分解为G=S×F×T,其中S= 这里d_i都是正整数,满足d_1|d_2|…|d_r,F是秩为s的自由Abel群,T是有限Abel群,T=Z_(e_1)⊕Z_(e_2)⊕…⊕Z_e_t,e_11,满足e_1|e_2|…|e_t,并且(d_1,e_t)=1.进一步,(d_1,d_2,…,d_T;s;e_1,e_2,…,e_t)是群G的同构不变量,即若群H也是无限循环群被有限生成Abel群的中心扩张,T_H是ζH的挠子群.如果T_H的阶与ζH/(H'⊕T_H)的挠子群的阶互索,那么G同构于H的充要条件是它们有相同的不变量.显然,这个结果涵盖了有限生成Abel群的结构定理.