Hyperbolic groups and their quotients of bounded exponents
S.
V.
Ivanov;
A.
Yu.
Ol'shanskii
2091-2138
Abstract: In 1987, Gromov conjectured that for every non-elementary hyperbolic group $G$ there is an $n =n(G)$ such that the quotient group $G/G^{n}$ is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of $G/G^{n}$ is given, it is proven that the word and conjugacy problem are solvable in $G/G^{n}$ and that $\bigcap _{k=1}^{\infty }G^{k} = \{ 1\}$. The proofs heavily depend upon prior authors' results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents.
An Application of Convex Integration to Contact Geometry
Hansjörg
Geiges;
Jesús
Gonzalo
2139-2149
Abstract: We prove that every closed, orientable $3$-manifold $M$ admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov's convex integration technique and the $h$-principle. Similar methods can be used to show that $M$ admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If $M$ is a closed $(2n+1)$-manifold with contact form $\omega$ whose contact distribution $\ker \omega$ admits $k$ everywhere linearly independent sections, then $M$ admits $k+1$ linearly independent contact forms with linearly independent Reeb vector fields.
Bott's vanishing theorem for regular Lie algebroids
Jan
Kubarski
2151-2167
Abstract: Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example: --- differential groupoids, --- principal bundles, --- vector bundles, --- actions of Lie groups on manifolds, --- transversally complete foliations, --- nonclosed Lie subgroups, --- Poisson manifolds, --- some complete closed pseudogroups. We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).
Cesàro Summability of Two-dimensional Walsh-Fourier Series
Ferenc
Weisz
2169-2181
Abstract: We introduce p-quasi-local operators and the two-dimensionaldyadic Hardy spaces $H_{p}$ defined by the dyadic squares. It is proved that, if a sublinear operator $T$ is p-quasi-local and bounded from $L_{\infty }$ to $L_{\infty }$, then it is also bounded from $H_{p}$ to $L_{p}$ $(0<p \leq 1)$. As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from $H_{p}$ to $L_{p}$ $(1/2<p \leq \infty )$ and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function $f \in L_{1}$ converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from $H_{p}$ to $L_{p}$ for every $0<p \leq \infty$.
$\Omega$-inverse limit stability theorem
Hiroshi
Ikeda
2183-2200
Abstract: We prove that if an endomorphism $f$ satisfies weak Axiom A and the no-cycles condition then $f$ is $\Omega$-inverse limit stable. This result is a generalization of Smale's $\Omega$-stability theorem from diffeomorphisms to endomorphisms.
Real connective K-theory and the quaternion group
Dilip
Bayen;
Robert
R.
Bruner
2201-2216
Abstract: Let $ko$ be the real connective K-theory spectrum. We compute $ko_*BG$ and $ko^*BG$ for groups $G$ whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients $t(ko)^G_*$ of the $G$ fixed points of the Tate spectrum $t(ko)$ for $G = Sl_2(3)$ and $G = Q_8$. The results provide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that $t(ko)^G$ is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.
Relatively free invariant algebras of finite reflection groups
Mátyás
Domokos
2217-2234
Abstract: Let $G$ be a finite subgroup of $Gl_{n}(K)$ $(K$ is a field of characteristic $0$ and $n\geq 2)$ acting by linear substitution on a relatively free algebra $K\langle x_{1},\hdots ,x_{n}\rangle /I$ of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if $G$ is a pseudo-reflection group and $I$ contains the polynomial $[[x_{2},x_{1}],x_{1}].$
The Structure and Enumeration of Link Projections
Martin
Bridgeman
2235-2248
Abstract: We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well-known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.
Special values of symmetric hypergeometric functions
Francesco
Baldassarri
2249-2289
Abstract: We discuss the $p$-adic formula (0.3) of P. Th. Young, in the framework of Dwork's theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it's not a priori clear that the Frobenius structure should extend to that disk. But the singularity is logarithmic, and it extends to a divisor with normal crossings relative to $\mathbf {Z}_{p}$ in $\mathbf {P}^{1}_{\mathbf {Z}_{p}}$. We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as a logarithmic $F$-subcrystal to the unit disk at 0. So, in these optics, ``singular classes are not supersingular''. If, in particular, the holomorphic solution at 0 is bounded, the extended logarithmic $F$-crystal has no singularity in the residue class of 0, so that it is an $F$-crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.
$C^*$-Algebras with the Approximate Positive Factorization Property
G.
J.
Murphy;
N.
C.
Phillips
2291-2306
Abstract: We say that a unital $\mathrm {C}^{*}$-algebra $A$ has the approximate positive factorization property (APFP) if every element of $A$ is a norm limit of products of positive elements of $A$. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of $\mathrm {C}^{*}$-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial $K_{1}$, and stable rank 1. In the unital case, the $K_{0}$ group separates the tracial states. The APFP passes to matrix algebras, and if $I$ is an ideal in $A$ such that $I$ and $A/I$ have the APFP, then so does $A$. We also give some new examples of $\mathrm {C}^{*}$-algebras with the APFP, including type $\mathrm {II}_{1}$ factors and infinite-dimensional simple unital direct limits of homogeneous $\mathrm {C}^{*}$-algebras with slow dimension growth, real rank zero, and trivial $K_{1}$ group. Simple direct limits of homogeneous $\mathrm {C}^{*}$-algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a $\mathrm {C}^{*}$-algebra that may be of interest in the future.
Boundary and Lens Rigidity of Lorentzian Surfaces
Lars
Andersson;
Mattias
Dahl;
Ralph
Howard
2307-2329
Abstract: Let $g$ be a Lorentzian metric on the plane $\r ^2$ that agrees with the standard metric $g_0=-dx^2+dy^2$ outside a compact set and so that there are no conjugate points along any time-like geodesic of $(\r ^2,g)$. Then $(\r ^2,g)$ and $(\r ^2,g_0)$ are isometric. Further, if $(M,g)$ and $(M^*,g^*)$ are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of $(M,g)$ maximize the distances between their points and $(M,g)$ and $(M^*,g^*)$ are ``boundary isometric'', then there is a conformal diffeomorphism between $(M,g)$ and $(M^*,g^*)$ and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.
Composition operators between Bergman and Hardy spaces
Wayne
Smith
2331-2348
Abstract: We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces.
Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities
C.
Rebelo;
F.
Zanolin
2349-2389
Abstract: We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation $x'' + g(x) = s + w(t,x)$, as $s\to +\infty$ or $s\to - \infty ,$ where $g$ is a smooth function defined on a open interval $]a,b[\subset {\mathbb {R}}.$ The hypotheses we assume on the nonlinearity $g(x)$ allow us to cover the case $b=+\infty$ (or $a = -\infty$) and $g$ having superlinear growth at infinity, as well as the case $b < +\infty$ (or $a > -\infty$) and $g$ having a singularity in $b$ (respectively in $a$). Applications are given also to situations like $g'(-\infty ) \not = g'(+\infty )$ (including the so-called ``jumping nonlinearities''). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.
Smooth classification of geometrically finite one-dimensional maps
Yunping
Jiang
2391-2412
Abstract: The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Hölder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete $C^{1}$-invariants within a mixing topological conjugacy class.
Total absolute curvature and tightness of noncompact manifolds
Martin
van Gemmeren
2413-2426
Abstract: In the first part we prove an extension of the Chern-Lashof inequality for noncompact immersed manifolds with finitely many ends. For this we give a lower bound of the total absolute curvature in terms of topological invariants of the manifold. In the second part we discuss tightness properties for such immersions. Finally, we give an upper bound for the substantial codimension.
Tensor products over abelian $W^*$-algebras
Bojan
Magajna
2427-2440
Abstract: Tensor products of C$^*-$algebras over an abelian W$^*-$algebra $Z$ are studied. The minimal C$^*-$norm on $A\odot _ZB$ is shown to be just the quotient of the minimal C$^*-$norm on $A\odot B$ if $A$ or $B$ is exact.
Fine structure of the space of spherical minimal immersions
Hillel
Gauchman;
Gabor
Toth
2441-2463
Abstract: The space of congruence classes of full spherical minimal immersions $f:S^m\to S^n$ of a given source dimension $m$ and algebraic degree $p$ is a compact convex body $\mathcal {M}_m^p$ in a representation space $\mathcal {F}_m^p$ of the special orthogonal group $SO(m+1)$. In Ann. of Math. 93 (1971), 43--62 DoCarmo and Wallach gave a lower bound for $\mathcal {F}_m^p$ and conjectured that the estimate was sharp. Toth resolved this ``exact dimension conjecture'' positively so that all irreducible components of $\mathcal {F}_m^p$ became known. The purpose of the present paper is to characterize each irreducible component $V$ of $\mathcal {F}_m^p$ in terms of the spherical minimal immersions represented by the slice $V\cap \mathcal {M}_m^p$. Using this geometric insight, the recent examples of DeTurck and Ziller are located within $\mathcal {M}_m^p$.
Prime spectra of quantum semisimple groups
K.
A.
Brown;
K.
R.
Goodearl
2465-2502
Abstract: We study the prime ideal spaces of the quantized function algebras $R_{q}[G]$, for $G$ a semisimple Lie group and $q$ an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras $R_{q}[G]$ satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar's strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature -- it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of $R_{q}[G]$. In the final section the results are specialized to the case $G= SL_{n}(\mathbb {C})$, where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine $n$-space satisfy our axiom scheme when the group generated by the parameters is torsionfree.
Hardy spaces and twisted sectors for geometric models
Pietro
Poggi-Corradini
2503-2518
Abstract: We study the one-to-one analytic maps $\sigma$ that send the unit disc into a region $G$ with the property that $\lambda G\subset G$ for some complex number $\lambda$, $0<|\lambda |<1$. These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region $G$ that characterize their rate of growth, i.e. we prove that $\sigma \in\bigcap _{p<\infty }H^p$ if and only if $G$ does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.