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1.
Irine Peng 《Geometric And Functional Analysis》2011,21(4):951-1000
We show quasi-isometric rigidity for a class of finitely generated, non-polycyclic nilpotent-by-cyclic groups. Specifically,
let Γ1, Γ2 be ascending HNN extensions of finitely generated nilpotent groups N
1 and N
2, such that Γ1 is irreducible (see Definition 1.1). If Γ1 and Γ2 are quasi-isometric to each other then N
1 and N
2 are virtual lattices in a common simply connected nilpotent Lie group [(N)\tilde]{\tilde{N}}. As a consequence, we show the class of irreducible ascending HNN extensions of finitely generated nilpotent groups is quasi-isometrically
rigid. 相似文献
2.
Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely
generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime
inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal
class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal
class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings
between members of each Mislin genus. In this sense, the family Γ resembles the family N0 of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with
N0, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes.
We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent
groups of class 2 nor for torsion-free abelian groups of finite rank.
Supported by DGICYT grant PB94-0725
This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag. 相似文献
3.
Berthold J. Maier 《Israel Journal of Mathematics》1983,46(3):170-188
Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ
+, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup
of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N
0.
Some results of this paper were contained in [6]. 相似文献
4.
Gemma Parmeggiani 《Rendiconti del Circolo Matematico di Palermo》1995,44(2):215-238
In this paper we investigate the class of finite soluble groups in which every subnormal subgroup has normal normalizer. In
particular we prove that they areUN
2U, whereU andN
2denote finite abelian groups and of finite nilpotent groups of class at most 2 respectively. 相似文献
5.
Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? K (G), grad(?)(? K (G)), grad(g)(exp ? K (G)), and L K (G). Let 𝔗 c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F n (𝔗 c ) be the relatively free group of rank n in 𝔗 c . We prove that ? K (F n (𝔗 c )) is relatively free in some variety of nilpotent Lie algebras, and ? K (F n (𝔗 c )) ? L K (F n (𝔗 c )) ? grad(?)(? K (F n (𝔗 c ))) ? grad(g)(exp ? K (F n (𝔗 c ))) as Lie algebras in a natural way. Furthermore, F n (𝔗 c ) is a Magnus nilpotent group. Let G 1 and G 2 be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G 1 and G 2 are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ??(H) ? L ?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? K and L K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ??(G) is not isomorphic to L ?(G) as Lie algebras. 相似文献
6.
We find the nilpotency class of a group of 2-symmetric words for free nilpotent groups, free nilpotent metabelian groups, and free (nilpotent of class c)-by-Abelian groups. 相似文献
7.
《Quaestiones Mathematicae》2013,36(1-3):143-155
We study the question of what properties of nilpotent groups are shared by their abelianizations. We identify two such properties—that of being a π-torsion group, where π is a family of primes, and that of having qth roots, for some prime q. We use these properties to provide simplified proofs of the following theorems in the localization of nilpotent groups. Let H, K be subgroups of the nilpotent group N and let P be a family of primes. Then [H, K] P = [HP, Kp] Let the group G act on the nilpotent group N. Then G acts compatibly on Np and (ΓG i N)P = ΓG i(Np). The second theorem above is then applied to the study of the localization of relative groups, in the sense of [4]. 相似文献
8.
The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M p , N p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N 相似文献
9.
D. Saracino 《Israel Journal of Mathematics》1976,25(3-4):241-248
Letn be a positive integer, letK
n denote the theory of groups nilpotent of class at mostn, and letK
n
+
denote the theory of torsion-free groups nilpotent of class at mostn. We show that ifn≧2 then neitherK
n norK
n
+
has a model companion. ForK
n we obtain the stronger result that the class of finitely generic models is disjoint from the class of infinitely generic
models. We also give some other results about existentially complete nilpotent groups.
Dedicated to the Memory of Abraham Robinson. 相似文献
10.
H. J. Rivertz 《Ukrainian Mathematical Journal》2009,61(12):1946-1955
In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant
is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on
some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary
leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general
class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q
c
4. 相似文献
11.
V. Roman’kov 《Journal of Pure and Applied Algebra》2011,215(4):664-671
Let N be a finitely generated nilpotent group. We show that there is an algorithm that for any automorphism φ∈Aut(N) computes its Reidemeister number R(φ). It is proved that any free nilpotent group Nrc of rank r and class c belongs to class R∞ if any of the following conditions holds: r=2 and c≥4; r=3 and c≥12; r≥4 and c≥2r. 相似文献
12.
Rainer Felix 《Monatshefte für Mathematik》1982,94(2):91-101
We contribute some results to the following question discussed in several papers ([3], [9], [12]): To what extent the characters of the irreducible representations of a nilpotent Lie groupN determine all central tempered distributions onN. A hereditary property is proved, which enables us to give a negative answer for a large class of nilpotent Lie groups. However, we give a positive answer for a certain series of two-step nilpotent groups. Furthermore, proving a generalisation of the Schwartz Kernel Theorem we decide the question specially for all groups of dimension<5. 相似文献
13.
Lisa DeMeyer 《manuscripta mathematica》2001,105(3):283-310
We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice
in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie
groups N which are constructed using irreducible representations of the compact Lie group SU(2).
Received: 8 November 2000 / Revised version: 9 April 2001 相似文献
14.
We define a partial order on the set No,c of pairs (O,C), where O is a nilpotent orbit and C is a conjugacy class in A(O), Lusztig’s canonical quotient of A(O). We
then show that there is a unique order-reversing duality map No,c → LNo,c that has certain properties analogous to those of the original Lusztig–Spaltenstein duality map. This generalizes work of
E. Sommers. 相似文献
15.
Zahedeh Azhdari 《代数通讯》2013,41(10):4133-4139
Let G be a group and Autc(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Autc(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* = Autc(G). In this article, we characterize all finitely generated groups G for which the equality Autc(G) = Inn(G) holds. 相似文献
16.
Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G2 structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained
in G2. In the process we also find a new metric with exceptional holonomy.
Received: 20 September 相似文献
17.
Primož Moravec 《Semigroup Forum》2008,77(2):316-324
In this paper we find simple characterizations of completely simple semigroups with H-classes nilpotent of class ≤c, and of completely simple semigroups whose core has H-classes nilpotent of class ≤c. The notion of w-marginal completely regular semigroups is introduced, generalizing the concept of central semigroups. A law characterizing
[x
1,x
2,…,x
c+1]-marginal completely simple semigroups is obtained. Additionally, the least congruences corresponding to these classes are
described. Our results extend the corresponding results obtained by Petrich and Reilly in the abelian case.
The author was supported by the Ministry of Higher Education, Science and Technology of Slovenia. 相似文献
18.
Constantinos E. Kofinas 《代数通讯》2013,41(4):1575-1593
Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL n (?) and N Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras. 相似文献
19.
It is known that the Mislin genus of a finitely generated nilpotent group N with finite commutator subgroup admits an abelian group structure. In this paper, we compute explicitly that structure under the following additional assumptions: The torsion subgroup TN is abelian, the epimorphism N→N/TN splits and all automorphisms of TN commute with cinjugation by elements of N. Among the groups satisfying these conditions are all nilpotent split extensions of a finite cyclic group by a finitely free abelian group. We further prove that the function M ? M × Nk1 k ≥ 2, which is in general a surjective homomorphism from the genus of N onto the genus of Nk , is an isomorphism at least in an imporatnt special case. Applications to the study of non-cancellation phenomena in group theory are given. 相似文献
20.
We will say that a subgroup X of G satisfies property C in G if CG(X?Xg)\leqq X?Xg{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then CG(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group. 相似文献