Free actions of -groups on products of lens spaces

Let be an odd prime number. We prove that if acts freely on a product of equidimensional lens spaces, then . This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian -groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A -group can act freely on a product of lens spaces with a trivial action on homology if and only if and has the -extension property. The main technique is to study group extensions associated to free actions.

     

Distinguishing labeling of group actions

Suppose Γ is a group acting on a set X. An r-labeling f:X→{1,2,…,r} of X is distinguishing (with respect to Γ) if the only label preserving permutation of X in Γ is the identity. The distinguishing number, D_Γ(X), of the action of Γ on X is the minimum r for which there is an r-labeling which is distinguishing. This paper investigates the relation between the cardinality of a set X and the distinguishing numbers of group actions on X. For a positive integer n, let D(n) be the set of distinguishing numbers of transitive group actions on a set X of cardinality n, i.e., D(n)={D_Γ(X):|X|=n and Γ acts transitively on X}. We prove that . Then we consider the problem of an arbitrary fixed group Γ acting on a large set. We prove that if for any action of Γ on a set Y, for each proper normal subgroup H of Γ, D_H(Y)≤2, then there is an integer n such that for any set X with |X|≥n, for any action of Γ on X with no fixed points, D_Γ(X)≤2.     

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

We study Lie group structures on groups of the form C ^∞(M, K), where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these results for the group of holomorphic maps on a complex manifold with values in a complex Lie group K. We further show that there exists a natural Lie group structure on if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.      

Periodic manifolds with spectral gaps

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number N we construct periodic manifolds such that the essential spectrum of the corresponding Laplacian has at least N open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.     

Cantor sets and numbers with restricted partial quotients

For let be a Cantor set constructed from the interval , and let . We derive conditions under which

When these conditions do not hold, we derive a lower bound for the Hausdorff dimension of the above sum and product. We use these results to make corresponding statements about the sum and product of sets , where is a set of positive integers and is the set of real numbers such that all partial quotients of , except possibly the first, are members of .

     

Bounded automorphisms of surfaces

The equivalence among bounded homeomorphisms of a surface, reducible homeomorphisms of the surface, bounded automorphisms of the surface group, induced or exchange automorphisms of the surface group, and the stabilizer of the translation length function of the tree action is established.     

Generalized group actions in a global setting

We study generalized group actions on differentiable manifolds in the Colombeau framework, extending previous work on flows of generalized vector fields and symmetry group analysis of generalized solutions. As an application, we analyze group invariant generalized functions in this setting.     

Investigations of isotropy and homogeneity of spacetime in first-order logic

We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in certain cases, homogeneity of time. In turn, homogeneity of time implies homogeneity of space in general, and the converse also holds true in certain cases.An important innovation in our approach is that formulations of physical properties are simultaneously empirical and axiomatic (in the sense of first-order mathematical logic). In this case, for example, rather than presuppose the existence of spacetime metrics – together with all the continuity and smoothness apparatus that would entail – the basic logical formulas underpinning our work refer instead to the sets of (idealised) experiments that support the properties in question, e.g., isotropy is axiomatised by considering a set of experiments whose outcomes remain unchanged under spatial rotation. Higher-order constructs are not needed.     

Actions of discrete groups on spheres and real projective spaces

In this paper, we first define discrete, smooth actions on , whose limit sets are Cantor sets wildly embedded in (Antoine's necklaces). Secondly, we define Schottky groups on real projective spaces of odd dimensions, . We lift these actions to (locally) projective actions on the sphere and consider the quotient space of the domain of discontinuity by the group to obtain new examples of manifolds with real projective structures. *This work was partially supported by PROMEP (SEP). **This work was partially supported by CONACyT (México), grant U1 55084, PAPIIT (UNAM) grant IN102108.     

On Riemannian coverings of manifolds admitting locally splitting actions

By studying the structure of certain covering manifolds, associated with a series of subgroups of ₁(M), we approximate the structure of the Riemannian manifoldM, provided that it admits a locally splitting action. The above series arises because of the local splitting of the action and the study is carried out via initial data. Also, we indicate how manifolds without conjugate points can be investigated using locally splitting actions of abelian Lie groups.     

Groups acting transitively on compact CR manifolds of hypersurface type

Let be a compact homogeneous manifold with acting effectively and with a -invariant CR structure of hypersurface type; then any maximal compact subgroup acts transitively on .

     

Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

In the study of the collapsed manifolds with bounded sectional curvature,the following two results provide basic tools:a(singular)fibration theorem by K.Fukaya[J.Differential Gcom.,1987.25(1):139156]and J.Cheeger,K.Fukaya,and M.Gromov[J.Airier.Math.Soc.,1992.5(2):327372],and the stability for isometric compact Lie group actions on manifolds by R.S.Palais[Bull.Amer.Math.Soc.,1961,67(4):362364]and K.Grove and H.Karcher[Math.Z..1973,132:1120].The main results in this paper(partially)generalize the two results to manifolds with local bounded Ricci covering geometry.     

On Growth Types of Quotients of Coxeter Groups by Parabolic Subgroups

The principal objects studied in this note are infinite, non-affine Coxeter groups W. A well-known result of de la Harpe asserts that such groups have exponential growth. We study the growth type of quotients of W by parabolic subgroups and by a certain class of reflection subgroups. Our main result is that these quotients have exponential growth as well.     

Category of G-Groups and its Spectral Category

Let G be a group. We analyse some aspects of the category G-Grp of G-groups. In particular, we show that a construction similar to the construction of the spectral category, due to Gabriel and Oberst, and its dual, due to the second author, is possible for the category G-Grp.     

A counterexample to questions on the integrality property of virtual signature

Wall (1961), defined the virtual Euler Characteristic χ(Γ) of an arbitrary group Γ of finite homological type as , where Γ′ is any torsion free subgroup of finite index in Γ. Analogous to virtual Euler Characteristic, we define the Virtual signature of an oriented virtual Poincare Duality group, a rational number. We show that two of Ken Brown's results on questions regarding the integrality property of virtual Euler Characteristics when formulated in the Virtual signature case is false.     

Topological,affine and isometric actions on flat riemannian manifolds II

Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π₀(Homeo(M))→Out π₁(M) can be lifted back to Homeo(M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.