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1.
Pierre De La Harpe 《Geometriae Dedicata》2002,95(1):1-17
This is an exposition of examples and classes of finitely-generated groups which have uniform exponential growth. The main examples are non-Abelian free groups, semi-direct products of free Abelian groups with automorphisms having an eigenvalue of modulus distinct from 1, and Golod–Shafarevich infinite finitely-generated p-groups. The classes include groups which virtually have non-Abelian free quotients, nonelementary hyperbolic groups, appropriate free products with amalgamation, HNN-extensions and one-relator groups, as well as soluble groups of exponential growth. Several open problems are formulated. 相似文献
2.
M. I. Hartley 《Discrete and Computational Geometry》1999,21(2):289-298
In this paper it is shown that any (abstract) polytope is a quotient of a regular polytope by some subgroup N of the automorphism group W of , and also that isomorphic polytopes are quotients of by conjugate subgroups of W . This extends work published in 1980 by Kato, who proved these results for a restricted class of polytopes which he called
``regular'. The methods used here are more elementary, and treat the problem in a purely nongeometric manner.
Received January 27, 1997, and in revised form October 1, 1997. 相似文献
3.
John R. Stembridge 《Journal of Algebraic Combinatorics》1996,5(4):353-385
Let W be a Coxeter group. We define an element w ε W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many
fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications
of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat
ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat
ordering coincide.
Partially supported by NSF Grants DMS-9057192 and DMS-9401575. 相似文献
4.
This paper studies free quotients of the groups SL2(ℤ[x]) and SL2(k[x, y]),k a finite field. These quotients give information about the relation of the above groups to their subgroups generated by elementary
or unipotent elements. 相似文献
5.
Michael Gass 《Semigroup Forum》1991,43(1):83-92
The compact subsets of a topological groupG form a semigroup,S(G), when multiplication is defined by set product. This semigroup is a topological semigroup when given the Vietoris topology.
It would be expected that the subgroups ofS(G) should in some way be related to the groupG. This is the case. It is shown that the subgroups ofS(G) are both algebraically and topologically exactly the groups obtained as quotients of certain subgroups ofG. One consequence of this is that every subgroup ofS(G) is a topological group. Conditions are also given for these subgroups to be open or closed.
Green's relations inS(G) have a particularly nice formulation. As a result, the relationsD andJ are equal inS(G). Moreover, the Schützenberger group of aD-class is a topological group that is topologically isomorphic to a quotient of certain subgroups ofG. 相似文献
6.
7.
Denis V. Osin 《Inventiones Mathematicae》2007,167(2):295-326
In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group
G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π1(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that
peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings.
Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26 相似文献
8.
In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation
subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups
and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and
define a group τ = [Σ(V×W⋃ *), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb
group lies in the center of τ, thereby improving a result of Varadarajan.
__________
Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 320–328, March, 2005. 相似文献
9.
Fanny Kassel 《Mathematische Annalen》2012,353(2):599-632
Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of standard compact quotients of G/H, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Γ keeps acting properly discontinuously and cocompactly on G/H. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n, 2)/U(n, 1) by Zariski-dense discrete subgroups of SO(2n, 2) acting properly discontinuously. 相似文献
10.
Todor Tsankov 《Geometric And Functional Analysis》2012,22(2):528-555
We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ??, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand?CRaikov theorem holds for topological subgroups of S ??: for all such groups, continuous irreducible representations separate points in the group. 相似文献
11.
A. D. Tavadze 《Journal of Mathematical Sciences》2008,155(5):670-696
By analogy with the notion of a pronilpotent W-group, we define the notion of a pronilpotent Lie algebra and establish a one-to-one correspondence between pronilpotent
W-groups and pronilpotent Lie algebras in the case where W is a field of zero characteristic; also, we establish a connection between free and projective groups of a given manifold.
We prove that, for some manifolds, free and projective groups coincide; we investigate conditions under which a subgroup is
free in an absolutely free pronilpotent W-group. In the class of pronilpotent groups, we introduce and discuss the notion of a free product; we construct an example
which shows that an analog of the Kurosh theorem on subgroups of a free product does not hold even for finitely generated
subgroups. A series of results stated in this paper was announced in [35–38].
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 49, Algebra
and Geometry, 2007. 相似文献
12.
Jonathan Brown 《Transformation Groups》2009,14(1):87-114
We construct an explicit set of generators for the finite W-algebras associated to nilpotent matrices in the symplectic or orthogonal Lie algebras whose Jordan blocks are all of the
same size. We use these generators to show that such finite W-algebras are quotients of twisted Yangians. 相似文献
13.
Koji Nuida 《代数通讯》2013,41(7):2559-2595
In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components. 相似文献
14.
A Lie group is called exponential if its exponential map is surjective. It is called weakly exponential if the exponential
image is dense, which is equivalent to the connectivity of each of the Cartan subgroups (compare [11]). In the present paper
the authors study exponential Lie groups which are of mixed type, i.e., neither solvable nor semisimple. Necessary conditions
and also, for special mixed Lie groups, sufficient conditions are given for exponentiality. Several counter examples are provided
showing that none of the conditions which have surfaced during the course of our investigation can work as necessary and sufficient ones. All conditions considered deal with centralizers of ad-nilpotent elements of the Lie algebra. For example,
it is shown that if G is exponential, there is a rather large characteristic subgroup B which contains the nilradical, all Levi factors, and all maximal compactly embedded subgroups, which is also exponential.
Moreover, this subgroup is also Mal’cev splittable so that one can apply earlier results on Mal’cev splittable exponential
Lie groups, which characterize exponentiality of these Lie groups (also by conditions concerning the centralizers of ad-nilpotent
elements).
(Received 1 June 1999; in final form 28 December 1999) 相似文献
15.
We investigate the imaginary cone in hyperbolic Coxeter systems in order to show that any Coxeter system contains universal reflection subgroups of arbitrarily large rank. Furthermore, in the hyperbolic case, the positive spans of the simple roots of the universal reflection subgroups are shown to approximate the imaginary cone (using an appropriate topology on the set of roots), answering a question due to Dyer [9] in the special case of hyperbolic Coxeter systems. Finally, we discuss growth in Coxeter systems and utilize the previous results to extend the results of [16] regarding exponential growth in parabolic quotients in Coxeter groups. 相似文献
16.
Luisa Paoluzzi 《代数通讯》2013,41(3):759-778
We classify quotients of type PSL(2,q) and PGL(2,q) with torsion-free kernel for four of the nine hyperbolic tetrahedral groups. Using this result, we give a classification of the quotients with torsion-free kernel of type PSL(2q) ×Z2 of the associated Coxeter or reflection groups. These do not admit quotients of type PSL(2,q),PGL(2,q). We also study quotients of type PSL(2,q) and PGL(2,q) of the fundamental group of the hyperbolic 3-orbifold of minimal known volume. 相似文献
17.
C.K. Fan 《Journal of Algebraic Combinatorics》1996,5(3):175-189
Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \-H of H and show that it has a basis parametrized by a certain subset W
cof the Coxeter group W. Specifically, W
cconsists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W
cand compute the cardinality of W
cwhen W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w W
cof an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \-H. 相似文献
18.
Gregor Kemper 《manuscripta mathematica》1996,90(1):343-363
A general method is developed to attack Noether's Problem constructively by trying to find minimal bases consisting of rational
invariants which are quotients of polynomials of small degrees. This approach turns out to be successful for many small groups
and for most of the classical groups with their natural representations. The applications include affirmative answers to Noether's
Problem for the conformal symplectic groups CSp
2n
(q), for the simple subgroups Ω
n
(q) of the orthogonal groups forn andq odd, for some other subgroups of orthogonal groups and for the special unitary groups SU
n
(q
2).
The author was supported by the Graduate College “Modelling and Scientific Computing in Mathematics and Science” during this
work 相似文献
19.
Zoran Šunić 《Archiv der Mathematik》2009,93(1):23-28
Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms
of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic
and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on
the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H
i
} of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H
i
have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union
generates G can be replaced by the condition that it is equal to G).
The work presented here is partially supported by NSF/DMS-0805932. 相似文献
20.
We determine the order dimension of the strong Bruhat order on finite Coxeter groups of types A, B and H. The order dimension is determined using a generalization of a theorem of Dilworth: dim (P)=width(Irr(P)), whenever P satisfies a simple order-theoretic condition called here the dissective property (or clivage). The result for dissective posets follows from an upper bound and lower bound on the dimension of any finite poset. The dissective property is related, via MacNeille completion, to the distributive property of lattices. We show a similar connection between quotients of the strong Bruhat order with respect to parabolic subgroups and lattice quotients. 相似文献