Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .      

Penalty functions in subanalytic optimization

We define an exact penalty function for optimization problems defined by subanalytic functions with equality or inequality constraints     

Cremona transformations and diffeomorphisms of surfaces

We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.     

On circle diffeomorphisms with discontinuous derivatives and quasi-invariance subgroups of Malliavin-Shavgulidze measures

This note is devoted to construction of the domain of the regular representation of Diff₊(S¹). We consider a subgroup of diffeomorphism group such that the Malliavin-Shavgulidze measure is quasi-invariant with respect to the left action of the subgroup. The measure appears to be quasi-invariant with respect to the left action of diffeomorphisms with discontinuous second derivative. We derive an expression for quasi-invariance density and obtain an additional PSL(2,R) symmetry breaking term.     

On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit

It is known that the equivariant diffeomorphism group Diff_G^∞(M)₀ of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of Diff_G^∞(M)₀ when M is a G-manifold with codimension one orbit.     

A nonsmooth Morse-Sard theorem for subanalytic functions

According to the Morse-Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse-Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the functions under consideration to be semialgebraic or subanalytic. We make no assumption of subdifferential regularity. ?ojasiewicz-type inequalities for nonsmooth functions follow quickly from tools of the kind we develop, leading to convergence theory for subgradient dynamical systems.     

Weighted diffeomorphism groups of Riemannian manifolds

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that for the special case where the manifold is the euclidean space, these Lie groups coincide with the ones constructed in the author’s earlier work (Walter, 2012).     

Finite determinacy and polynomial normal forms for diffeomorphisms near a strongly 1-resonant fixed point

In this paper we study polynomial normal forms and smooth (C^∞$C^{\infty }$) classification of one kind of diffeomorphisms which have infinite many resonant relations but are finitely determined. We derive a complete list of normal forms of all such germs with arbitrary degeneracy of their nonlinear parts.     

The Birkhoff decomposition in groups of formal diffeomorphisms

Let $G_{\infty }$ be the group of one parameter identity-tangent diffeomorphisms on the line whose coefficients are formal Laurent series in the parameter ε with a pole of finite order at 0. It is well-known that the Birkhoff decomposition can be defined in such a group. We investigate the stability of the Birkhoff decomposition in subgroups of $G_{\infty }$ and give a formula for this decomposition. As proven by A. Connes and D. Kreimer, the Birkhoff decomposition is related to renormalization in quantum field theory and we give an application of our results in the last section. To cite this article: F. Menous, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Harborth constants for certain classes of metacyclic groups

The Harborth constant of a finite group $G$ is the smallest integer $k\ge exp\left(G\right)$ such that any subset of $G$ of size $k$ contains $exp\left(G\right)$ distinct elements whose product is 1. Generalizing previous work on the Harborth constants of dihedral groups, we compute the Harborth constants for the metacyclic groups of the form $H_{n,m}=〈x,y\mid x^n=1,y^2=x^m,yx=x^{?1}y〉$. We also solve the “inverse” problem of characterizing all smaller subsets that do not contain $exp\left(H_{n,m}\right)$ distinct elements whose product is 1.     

On the Tate-Shafarevich groups of certain elliptic curves

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.     

The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic groups and their subgroups. Such groups are called finitedimensional. We obtain that cyclic groups, virtually polycyclic groups, finitely generated virtually nilpotent groups and connected Lie groups of local biholomorphisms are finite-dimensional. We provide several methods to identify finite-dimensional groups and build examples.As a consequence we generalize results of Arnold, Seigal–Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups and in particular finitely generated virtually nilpotent groups.     

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups and with n a suitable power of a prime number p.     

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D _infw,0 ^supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D _infw,0 ^supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program     

Lawlessness and rank restrictions in certain finitary groups

We give two applications of the recent classification of locally finite simple finitary skew linear groups. We show that certain irreducible finitary skew linear groups of infinite dimension generate the variety of all groups and have infinite Prüfer rank.

     

The Lappo-Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.