On mapping class groups of contractible open $3$-manifolds
Robert
Myers
1-46
Abstract: Let $W$ be an irreducible, eventually end-irreducible contractible open $3$-manifold other than ${{\mathbf{R}}^3}$, and let $V$ be a "good" exhaustion of $ W$. Let $\mathcal{H}(W;V)$ be the subgroup of the mapping class group $ \mathcal{H}(W)$ which is "eventually carried by $V$." This paper shows how to compute $\mathcal{H}(W;V)$ in terms of the mapping class groups of certain compact $3$-manifolds associated to $V$. The computation is carried out for a genus two example and for the classical genus one example of Whitehead. For these examples $\mathcal{H}(W) = \mathcal{H}(W;V)$.
Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $3$-space
Fritz
Grunewald;
Joachim
Schwermer
47-78
Abstract: Let $\mathcal{O}$ be the ring of integers in an imaginary quadratic number-field. The group $ {\text{PSL}}_2(\mathcal{O})$ acts discontinuously on hyperbolic $3$-space $H$. If $\Gamma \leq {\text{PSL}}_2(\mathcal{O})$ is a torsionfree subgroup of finite index then the manifold $ \Gamma \backslash H$ can be compactified to a manifold $ {M_\Gamma }$ so that the inclusion $\Gamma \backslash H \leq {M_\Gamma }$ is a homotopy equivalence. $ {M_\Gamma }$ is a compact with boundary. The boundary being a union of finitely many $2$-tori. This paper contains a computer-aided study of subgroups of low index in $ {\text{PSL}}_2(\mathcal{O})$ for various $ \mathcal{O}$. The explicit description of these subgroups leads to a study of the homeomorphism types of the ${M_\Gamma }$.
Pseudobases in direct powers of an algebra
Paul
Bankston
79-90
Abstract: A subset $ P$ of an abstract algebra $ A$ is a pseudobasis if every function from $P$ into $A$ extends uniquely to an endomorphism on $ A$. $A$ is called $\kappa$-free has a pseudobasis of cardinality $ \kappa$; $A$ is minimally free if $ A$ has a pseudobasis. (The 0-free algebras are "rigid" in the strong sense; the $ 1$-free groups are always abelian, and are precisely the additive groups of $ E$-rings.) Our interest here is in the existence of pseudobases in direct powers $ {A^I}$ of an algebra $ A$. On the positive side, if $A$ is a rigid division ring, $\kappa$ is a cardinal, and there is no measurable cardinal $\mu$ with $ \vert A\vert < \mu \leq \kappa$, then ${A^I}$ is $\kappa$-free whenever $\vert I\vert = \vert{A^\kappa }\vert$. On the negative side, if $A$ is a rigid division ring and there is a measurable cardinal $\mu$ with $ \vert A\vert < \mu \leq \vert I\vert$, then ${A^I}$ is not minimally free.
$K$-theory of Eilenberg-Mac Lane spaces and cell-like mapping problem
A. N.
Dranishnikov
91-103
Abstract: There exist cell-like dimension raising maps of $6$-dimensional manifolds. The existence of such maps is proved by using $K$-theory of Eilenberg-Mac Lane complexes.
Toral actions on $4$-manifolds and their classifications
M. Ho
Kim
105-130
Abstract: The existence of a cross-section is proved for some nonorientable $ 4$-manifolds with a $ {T^2}$-action. Two $ 4$-manifolds with a $ {T^2}$-action, which have the same previously known invariants, are constructed. By using a new homotopy invariant, they are proved to be homotopy inequivalent. Finally a stable diffeomorphism theorem is proved.
The semigroup property of value functions in Lagrange problems
Peter R.
Wolenski
131-154
Abstract: The Lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form. The binary operation of inf-composition applied to the value functions of a Lagrange problem equates the principle of optimality with a semigroup property. This paper finds the infinitesimal generator of the semigroup by differentiating at $t = 0$. The type of limit is epigraphical convergence in a uniform sense. Moreover, the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed. The main results provide a new approach to existence and uniqueness questions in Hamilton-Jacobi theory. When $ L$ is in addition finite-valued, the results are given in terms of pointwise convergence.
Extensions of \'etale by connected group spaces
David B.
Jaffe
155-173
Abstract: The main theorem, in rough terms, asserts the following. Let $ K$ and $D$ be group spaces over a scheme $ S$. Assume that $ K$ has connected fibers and that $D$ is finite and étale over $ S$ . Assume that there exists a single finite, surjective, étale, Galois morphism $ \overline S \to S$ which decomposes (scheme-theoretically) every extension of $D$ by $K$. Let $\pi = \operatorname{Aut}(\overline S /S)$. Then group space extensions of $D$ with kernel $K$ are in bijective correspondence with pairs $(\xi ,\chi )$ consisting of a $ \pi$-group extension $\displaystyle \xi :1 \to K(\overline S) \to X \to D(\overline S ) \to 1$ and a $\pi$-group homomorphism $\chi :X \to \operatorname{Aut}(\overline K )$ which lifts the conjugation map $X \to \operatorname{Aut}(K(\overline S ))$ and which agrees with the conjugation map $K(\bar S) \to \operatorname{Aut}(\overline K )$. In this way, the calculation of group space extensions is reduced to a purely group-theoretic calculation.
Obstructions and hypersurface sections (minimally elliptic singularities)
Kurt
Behnke;
Jan Arthur
Christophersen
175-193
Abstract: We study the obstruction space ${T^2}$ for minimally elliptic surface singularities. We apply the main lemma of our previous paper [3] which relates ${T^2}$ to deformations of hypersurface sections. To use this we classify general hypersurface sections of minimally elliptic singularities. As in the rational singularity case there is a simple formula for the minimal number of generators for ${T^2}$ as a module over the local ring. This number is in many cases (e.g. for cusps of Hilbert modular surfaces) equal to the vector space dimension of ${T^2}$.
Intersection theory of linear embeddings
Sean
Keel
195-212
Abstract: We study intersection theoretic properties of subschemes defined by ideal sheaves of linear type in particular their behavior with respect to blowups and segre classes.
Functorial construction of Le Barz's triangle space with applications
Sean
Keel
213-229
Abstract: We give a new functorial construction of the space of triangles introduced by Le Barz. This description is used to exhibit the space as a composition of smooth blowups, to obtain a space of unordered triangles, and to study how the space varies in a family.
The first two obstructions to the freeness of arrangements
Sergey
Yuzvinsky
231-244
Abstract: In his previous paper the author characterized free arrangements by the vanishing of cohomology modules of a certain sheaf of graded modules over a polynomial ring. Thus the homogeneous components of these cohomology modules can be viewed as obstructions to the freeness of an arrangement. In this paper the first two obstructions are studied in detail. In particular the component of degree zero of the first nontrivial cohomology module has a close relation to formal arrangements and to the operation of truncation. This enables us to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement.
A dynamical proof of the multiplicative ergodic theorem
Peter
Walters
245-257
Abstract: We shall give a proof of the following result of Oseledec, in which $ GL(d)$ denotes the space of invertible $d \times d$ real matrices, $\vert\vert \bullet \vert\vert$ denotes any norm on the space of $d \times d$ matrices, and $ {\log ^+ }(t) = \max (0,\log (t))$ for $t \in [0,\infty )$.
Inverse monoids, trees and context-free languages
Stuart W.
Margolis;
John C.
Meakin
259-276
Abstract: This paper is concerned with a study of inverse monoids presented by a set $ X$ subject to relations of the form ${e_i} = {f_i}$, $i \in I$, where ${e_i}$ and ${f_i}$ are Dyck words, i.e. idempotents of the free inverse monoid on $X$. Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on $X$. In the finitely presented case the word problem is solved by using Rabin's theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored.
Quadratic transformation formulas for basic hypergeometric series
Mizan
Rahman;
Arun
Verma
277-302
Abstract: Starting with some of the known transformation formulas for well-poised $_2{\phi _1}$ and very-well-poised $_8{\phi _7}$ basic hypergeometric series we obtain $q$-analogues of $36$ quadratic transformation formulas given in $\S2.11$ of Higher transcendental functions, Vol. 1, edited by Erdélyi et al. We also derive some new quadratic transformation formulas that give rise to identities connecting very-well-poised but unbalanced $ _{10}{\phi _9}$ series in base $q$ with very-well-poised and balanced $_{12}{\phi _{11}}$ series in base $ {q^2}$. A Rogers-Ramanujan type identity is also found as a limiting case.
Harmonic volume, symmetric products, and the Abel-Jacobi map
William M.
Faucette
303-327
Abstract: The author generalizes B. Harris' definition of harmonic volume to the algebraic cycle ${W_k} - W_k^- $ for $k > 1$ in the Jacobian of a nonsingular algebraic curve $X$ . We define harmonic volume, determine its domain, and show that it is related to the image $ \nu$ of ${W_k} - W_k^-$ in the Griffiths intermediate Jacobian. We derive a formula expressing harmonic volume as a sum of integrals over a nested sequence of submanifolds of the $k$-fold symmetric product of $X$ . We show that $\nu$ , when applied to a certain class of forms, takes values in a discrete subgroup of $ {\mathbf{R}}/{\mathbf{Z}}$ and hence, when suitably extended to complexvalued forms, is identically zero modulo periods on primitive forms if $k \geq 2$. This implies that the image of ${W_k} - W_k^-$ is identically zero in the Griffiths intermediate Jacobian if $k \geq 2$. We introduce a new type of intermediate Jacobian which, like the Griffiths intermediate Jacobian, varies holomorphically with moduli, and we consider a holomorphic torus bundle on Torelli space with this fiber. We use the relationship mentioned above between $\nu$ and harmonic volume to compute the variation of $\nu$ when considered as a section of this bundle. This variational formula allows us to show that the image of ${W_k} - W_k^- $ in this intermediate Jacobian is nondegenerate.
A short proof of Zheludev's theorem
F.
Gesztesy;
B.
Simon
329-340
Abstract: We give a short proof of Zheludev's theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a Hill operator subject to certain short-range perturbations. As a by-product we simultaneously recover Rofe-Beketov's result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed Hill operator. Our methods are operator theoretic in nature and extend to other one-dimensional systems such as perturbed periodic Dirac operators and weakly perturbed second order finite difference operators. We employ the trick of using a selfadjoint Birman-Schwinger operator (even in cases where the perturbation changes sign), a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps.
Number of orbits of branch points of ${\bf R}$-trees
Renfang
Jiang
341-368
Abstract: An $R$-tree is a metric space in which any two points are joined by a unique arc. Every arc is isometric to a closed interval of $R$ . When a group $G$ acts on a tree ($Z$-tree) $X$ without inversion, then $X/G$ is a graph. One gets a presentation of $G$ from the quotient graph $X/G$ with vertex and edge stabilizers attached. For a general $R$-tree $X$, there is no such combinatorial structure on $ X/G$. But we can still ask what the maximum number of orbits of branch points of free actions on $R$-trees is. We prove the finiteness of the maximum number for a family of groups, which contains free products of free abelian groups and surface groups, and this family is closed under taking free products with amalgamation.
On Dehn functions and products of groups
Stephen G.
Brick
369-384
Abstract: If $G$ is a finitely presented group then its Dehn function--or its isoperimetric inequality--is of interest. For example, $G$ satisfies a linear isoperimetric inequality iff $ G$ is negatively curved (or hyperbolic in the sense of Gromov). Also, if $ G$ possesses an automatic structure then $G$ satisfies a quadratic isoperimetric inequality. We investigate the effect of certain natural operations on the Dehn function. We consider direct products, taking subgroups of finite index, free products, amalgamations, and HNN extensions.
Subvarieties of moduli space determined by finite groups acting on surfaces
John F. X.
Ries
385-406
Abstract: Suppose the finite group $G$ acts as orientation preserving homeomorphisms of the oriented surface $S$ of genus $g$. This determines an irreducible subvariety $ \mathcal{M}_g^{[G]}$ of the moduli space $ {\mathcal{M}_g}$ of Riemann surfaces of genus $g$ consisting of all surfaces with a group $ {G_1}$ of holomorphic homeomorphisms of the same topological type as $ G$. This family is determined by an equivalence class of epimorphisms $ \psi$ from a Fuchsian group $\Gamma$ to $G$ whose kernel is isomorphic to the fundamental group of $S$. To examine the singularity of ${\mathcal{M}_g}$ along this family one needs to know the full automorphism group of a generic element of $ \mathcal{M}_g^{[G]}$. In $ \S2$ we show how to compute this from $\psi$. Let $ \mathcal{M}_g^G$ denote the locus of all Riemann surfaces of genus $ g$ whose automorphism group contains a subgroup isomorphic to $G$. In $\S3$ we show that the irreducible components of this subvariety do not necessarily correspond to the families above, that is, the components cannot be put into a one-to-one correspondence with the topological actions of $G$. In $\S4$ we examine the actions of $G$ on the spaces of holomorphic $ k$-differentials and on the first homology. We show that when $G$ is not cyclic, the characters of these actions do not necessarily determine the topological type of the action of $G$ on $S$.
On the theory of Frobenius extensions and its application to Lie superalgebras
Allen D.
Bell;
Rolf
Farnsteiner
407-424
Abstract: By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory.
Actions of linearly reductive groups on PI-algebras
Nikolaus
Vonessen
425-442
Abstract: Let $G$ be a linearly reductive group acting rationally on a $ {\text{PI}}$-algebra $ R$. We study the relationship between $R$ and the fixed ring ${R^G}$ , generalizing earlier results obtained under the additional hypothesis that $R$ is affine.
A classification of the finite extensions of a multidimensional Bernoulli shift
Janet Whalen
Kammeyer
443-457
Abstract: The finite extensions of a multidimensional Bernoulli shift are classified completely, up to factor isomorphism, and up to isomorphism. If such an extension is weakly mixing then it must be Bernoulli; otherwise, it has a finite rotation factor, which has a Bernoulli complementary algebra. This result is extended to multidimensional Bernoulli flows and Bernoulli shifts of infinite entropy.