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1.
Let G be a finite abelian group. We list all the cases where the topological equivalence class of orientation-preserving free G-actions on a closed surface is unique. Moreover, we obtain the classification of topological equivalence classes of orientation-preserving free actions of finite abelian groups of rank 2. 相似文献
2.
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g?1 can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g−1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+?(G)|G|, where ?(G) is the maximal length of a chain of subgroups of G. There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained. 相似文献
3.
The concept of (stable) weak containment for measure-preserving actions of a countable group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points. 相似文献
4.
Olena Vaneeva 《Communications in Nonlinear Science & Numerical Simulation》2012,17(2):611-618
Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears to be a crucial point for the exhaustive solution of the problem. We prove that all the classes under consideration are normalized. This allows us to formulate the classification results in three ways: up to two kinds of equivalence (which are generated by transformations from the corresponding equivalence groups and all admissible point transformations) and using no equivalence. A simple way for the construction of exact solutions of mKdV-like equations using equivalence transformations is described. 相似文献
5.
Bruno Zimmermann 《Rendiconti del Circolo Matematico di Palermo》2003,52(1):47-56
We give a survey on large finite group actions on low-dimensional objects such as graphs, surfaces, handlebodies and closed
3-manifolds, and also on large finite subgroups of the outer automorphism groups of their fundamental groups (free groups
and surface groups) and of linear groups GL (n, ℤ); in particular we discuss recent results on maximal orders in various situations. 相似文献
6.
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip. 相似文献
7.
Bruno Zimmermann 《Rendiconti del Circolo Matematico di Palermo》1999,48(3):549-562
We study arbitrary (that is not necessarily orientation preserving) finite group actions on 3-dimensional orientable or nonorientable
handlebodies of genus g. For g>1, the maximal possible order is 24(g−1); we characterize the corresponding groups of this
order and also the occuring quotient orbifolds. Then we use this to study finite group actions of large order (with respect
to the equivariant Heegaard genus g) on closed 3-manifolds, again concentrating on the maximal case of order 24(g−1). Our
results extend corresponding results in the orientation preserving setting. Whereas for arbitrary finite group actions on
handlebodies much more types of quotient orbifolds occur than in the orientation preserving case, it turns out that for closed
3-manifolds the situation is quite rigid, in contrast to the orientation preserving case where one has many possibilities
to construct manifolds with large group actions. 相似文献
8.
本文主要考虑循环群作用 Riemann曲面的分类问题 ,我们列出了所有的循环群作用亏格为 4Riem ann曲面的拓扑分类和弱拓扑分类 相似文献
9.
Damien Gaboriau 《Journal of Functional Analysis》2011,260(2):414-427
We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S. Popa (2006) [Pop06, Def. 4.1]. There are continuum many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives, in particular, the first examples of such actions for the free group on 2 generators. 相似文献
10.
A finite group action on a lens space L(p,q) has ‘type OR’ if it reverses orientation and has an invariant Heegaard torus whose sides are interchanged by the orientation-reversing elements. In this paper we enumerate the actions of type OR up to equivalence. This leads to a complete classification of geometric finite group actions on amphicheiral lens spaces L(p,q) with p>2. The family of actions of type OR is partially ordered by lifting actions via covering maps. We show that each connected component of this poset may be described in terms of a subset of the lattice of Gaussian integers ordered by divisibility. This results in a correspondence equating equivalence classes of actions of type OR with pairs of Gaussian integers. 相似文献
11.
V. I. Berdyshev S. V. Emel’yanov A. B. Kurzhanski E. F. Mishchenko Yu. S. Osipov 《Proceedings of the Steklov Institute of Mathematics》2010,270(1):1-14
We consider the classification of germs of functions up to a nonstandard equivalence relation similar to the quasi boundary
equivalence and quasi equivalence of projections recently introduced by the second author. In fact, it is more rough than
the classification of functions with respect to the group of diffeomorphisms preserving a corner (that is, a union of a pair
of transversal hypersurfaces). We present the list of all simple classes and discuss its relation to the singularities of
Lagrangian projections with corners. Also, we describe the bifurcation diagrams and caustics of simple quasi corner singularities. 相似文献
12.
S. Allen Broughton 《Journal of Pure and Applied Algebra》2009,213(4):557-572
We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations. 相似文献
13.
R. Westwick 《Linear and Multilinear Algebra》1981,10(3):183-204
Complex trivectors have been classified up to rank 8 and real trivectors to rank 6. In this paper we determine the equivalence classes of real trivectors of rank 7. The equivalence relation is that induced by the general linear group on the underlying seven dimensional vector space. We have found three extra equivalence classes not in the list already present in the complex case and show that these complete the classification. 相似文献
14.
We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s classification of Serre subcategories. 相似文献
15.
An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G . We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group A5, and give various other examples. 相似文献
16.
The 3-coloring problem for a given graph consists in verifying whether it is possible to divide the vertex set of the graph into three subsets of pairwise nonadjacent vertices. A complete complexity classification is known for this problem for the hereditary classes defined by triples of forbidden induced subgraphs, each on at most 5 vertices. In this article, the quadruples of forbidden induced subgraphs is under consideration, each on atmost 5 vertices. For all but three corresponding hereditary classes, the computational status of the 3-coloring problem is determined. Considering two of the remaining three classes, we prove their polynomial equivalence and polynomial reducibility to the third class. 相似文献
17.
Lloyd Allison 《Journal of Algorithms in Cognition, Informatics and Logic》1981,2(3):227-244
A group of symmetries can divide the group of all permutations of n marks into equivalence classes of isomorphs; each class is a coset of the group of symmetries. An algorithm for generating a set of isomorph, or coset, representatives is discussed. It can be used when n! is too large for all permutations to be useful, but the number of equivalence classes is manageable. The group of symmetries can be varied to match many natural notions of similarity. A computation on permutations that is restricted to a set of representatives can then concentrate on the essentially different cases. 相似文献
18.
Courtney M. Thatcher 《Geometriae Dedicata》2010,148(1):391-415
In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same
odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product
as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions
and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis
and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently
vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined
by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions
and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal
invariants are computed in the process. 相似文献
19.
John D. Clemens 《Israel Journal of Mathematics》2009,170(1):113-123
We use the theory of Borel equivalence relations to analyze the equivalence relation of isomorphism among one-dimensional
subshifts. We show that this equivalence relation is a universal countable Borel equivalence relation, so that it admits no
definable complete invariants fundamentally simpler than the equivalence classes. We also see that the classification of higher
dimensional subshifts up to isomorphism has the same complexity as for the one-dimensional case. 相似文献
20.
We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris. 相似文献