On the asymptotic geometry of nonpositively curved graphmanifolds
S.
Buyalo;
V.
Schroeder
853-875
Abstract: In this paper we study the Tits geometry of a 3-dimensional graphmanifold of nonpositive curvature. In particular we give an optimal upper bound for the length of nonstandard components of the Tits metric. In the special case of a $\pi /2$-metric we determine the whole length spectrum of the nonstandard components.
Cyclic coverings and higher order embeddings of algebraic varieties
Thomas
Bauer;
Sandra
Di Rocco;
Tomasz
Szemberg
877-891
Abstract: In the present paper we study higher order embeddings in the context of cyclic coverings. Analyzing the positivity of the line bundle downstairs and its relationship with the branch divisor, we provide criteria for its pull-back to define an embedding of given order. We show that the obtained criteria are sharp. Finally, we apply them to various - sometimes seemingly unrelated-problems in algebraic geometry.
Bicanonical pencil of a determinantal Barlow surface
Yongnam
Lee
893-905
Abstract: In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.
Special values of multiple polylogarithms
Jonathan
M.
Borwein;
David
M.
Bradley;
David
J.
Broadhurst;
Petr
Lisonek
907-941
Abstract: Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.
Measuring the tameness of almost convex groups
Susan
Hermiller;
John
Meier
943-962
Abstract: A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points $x$ in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions.
A bounding question for almost flat manifolds
Shashidhar
Upadhyay
963-972
Abstract: We study bounding question for almost flat manifolds by looking at the equivalent description of them as infranilmanifolds $\Gamma\backslash L\rtimes G/G$. We show that infranilmanifolds $\Gamma\backslash L \rtimes G/G$ bound if $L$ is a 2-step nilpotent group and $G$ is finite cyclic and acts trivially on the center of the nilpotent Lie group $L$.
A model for the homotopy theory of homotopy theory
Charles
Rezk
973-1007
Abstract: We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models has a well-behaved internal hom-object.
Properties of Anick's spaces
Stephen
D.
Theriault
1009-1037
Abstract: We prove three useful properties of Anick's space $T^{2n-1}(p^{r})$. First, at odd primes a map from $P^{2n}(p^{r})$ into a homotopy commutative, homotopy associative $H$-space $X$ can be extended to a unique $H$-map from $T^{2n-1}(p^{r})$ into $X$. Second, at primes larger than $3$, $T^{2n-1}(p^{r})$ is itself homotopy commutative and homotopy associative. And third, the first two properties combine to show that the order of the identity map on $T^{2n-1}(p^{r})$ is $p^{r}$.
Transfers of Chern classes in BP-cohomology and Chow rings
Björn
Schuster;
Nobuaki
Yagita
1039-1054
Abstract: The $BP^*$-module structure of $BP^*(BG)$ for extraspecial $2$-groups is studied using transfer and Chern classes. These give rise to $p$-torsion elements in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained by Totaro.
A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $\mathbb{S}^n$
Mark
S.
Ashbaugh;
Rafael
D.
Benguria
1055-1087
Abstract: For a domain $\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\mathbb{S}^n$ we prove the optimal result $\lambda_2/\lambda_1(\Omega) \le \lambda_2/\lambda_1(\Omega^{\star})$ for the ratio of its first two Dirichlet eigenvalues where $\Omega^{\star}$, the symmetric rearrangement of $\Omega$ in $\mathbb{S}^n$, is a geodesic ball in $\mathbb{S}^n$ having the same $n$-volume as $\Omega$. We also show that $\lambda_2/\lambda_1$ for geodesic balls of geodesic radius $\theta_1$ less than or equal to $\pi/2$ is an increasing function of $\theta_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $\theta_1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $\theta_1=\pi/2$. Here $j_{\nu,k}$ denotes the $k$th positive zero of the Bessel function $J_{\nu}(t)$. This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of $\mathbb{S}^n$ and having a fixed value of $\lambda_1$ the one with the maximal value of $\lambda_2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $\lambda_2/\lambda_1$. Various other results for $\lambda_1$and $\lambda_2$ of geodesic balls in $\mathbb{S}^n$ are proved in the course of our work.
New range theorems for the dual Radon transform
Alexander
Katsevich
1089-1102
Abstract: Three new range theorems are established for the dual Radon transform $R^*$: on $C^\infty$ functions that do not decay fast at infinity (and admit an asymptotic expansion), on $\mathcal{S}(Z_n)$, and on $C_0^\infty(Z_n)$. Here $Z_n:=S^{n-1}\times\mathbb{R}$, and $R^*$ acts on even functions $\mu(\alpha,p)=\mu(-\alpha,-p), (\alpha,p)\in Z_n$.
Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations
Gui-Qiang
Chen;
Hermano
Frid
1103-1117
Abstract: We prove the uniqueness of Riemann solutions in the class of entropy solutions in $L^\infty\cap BV_{loc}$ for the $3\times 3$ system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global $L^2$-stability of the Riemann solutions even in the class of entropy solutions in $L^\infty$with arbitrarily large oscillation for the $3\times 3$ system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under $L^1$ perturbation of the Riemann initial data, as long as the corresponding solutions are in $L^\infty$ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions $U(x,t)$, piecewise Lipschitz in $x$, for any $t>0$.
Two-weight norm inequalities for Cesàro means of Laguerre expansions
Benjamin
Muckenhoupt;
David
W.
Webb
1119-1149
Abstract: Two-weight $L^{p}$ norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted'' cases, by including all values of $p\geq1$ for all positive orders of the Cesàro summation and all values of the Laguerre parameter $\alpha>-1$. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.
The FBI transform on compact ${\mathcal{C}^\infty}$ manifolds
Jared
Wunsch;
Maciej
Zworski
1151-1167
Abstract: We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact ${\mathcal{C}^\infty}$ manifold $M$. The FBI transform is a generalization of the classical notion of the wave-packet transform. We discuss the mapping properties of the FBI transform and its relationship to the calculus of pseudodifferential operators on $M$. We also describe the microlocal properties of its range in terms of the ``scattering calculus'' of pseudodifferential operators on the noncompact manifold $T^* M$.
Algebraic isomorphisms of limit algebras
A.
P.
Donsig;
T.
D.
Hudson;
E.
G.
Katsoulis
1169-1182
Abstract: We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.
Beyond Borcherds Lie algebras and inside
Stephen
Berman;
Elizabeth
Jurisich;
Shaobin
Tan
1183-1219
Abstract: We give a definition for a new class of Lie algebras by generators and relations which simultaneously generalize the Borcherds Lie algebras and the Slodowy G.I.M. Lie algebras. After proving these algebras are always subalgebras of Borcherds Lie algebras, as well as some other basic properties, we give a vertex operator representation for a factor of them. We need to develop a highly non-trivial generalization of the square length two cut off theorem of Goddard and Olive to do this.
Representations as elements in affine composition algebras
Pu
Zhang
1221-1249
Abstract: Let $A$ be the path algebra of a Euclidean quiver over a finite field $k$. The aim of this paper is to classify the modules $M$ with the property $[M]\in \mathcal{C}(A)$, where $\mathcal{C}(A)$ is Ringel's composition algebra. Namely, the main result says that if $\vert k\vert \ne 2, 3$, then $[M]\in \mathcal{C}(A)$ if and only if the regular direct summand of $M$ is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non-sincere. The main methods are representation theory of affine quivers, the structure of triangular decompositions of tame composition algebras, and the invariant subspaces of skew derivations. As an application, we see that $\mathcal{C}(A) = \mathcal{H}(A)$ if and only if the quiver of $A$is of Dynkin type.
The combinatorics of Bernstein functions
Thomas
J.
Haines
1251-1278
Abstract: A construction of Bernstein associates to each cocharacter of a split $p$-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz' conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of $Gl_n$; one example can be used to verify Kottwitz' conjecture for a special class of Shimura varieties (the ``Drinfeld case''). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the ``$\mu$-admissible'' set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.