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This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on C n \mathbb C^n . A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial f is smaller than expected. We carry out this study in the case that f is sufficiently non-degenerate with respect to its Newton polyhedron Γ(f), and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial f to the same question about polynomials f μ , where μ are faces of Γ(f) depending on the examined pole and f μ is obtained from f by throwing away all monomials of f whose exponents do not belong to μ. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial f μ , with μ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than expected.

We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any ∈ > 0 and any piece-wise affine map φ ∈ W 1,n (Ω; R n ) with |Dφ(x)| n ≤ Ldet Dφ(x) for almost every x ∈ Ω there exists a map u ∈ W 1,n (Ω; R n ) such that |Du(x)| n = L det Du(x) a.e. x ∈ Ω, u|∂Ω = φ, ∥u-φ∥L n (Ω) < ∈. The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Ježek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Ježek and McKenzie have “almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.

This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on R \mathbf R with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in D i f f 2 ( R ) \mathrm {Diff}^2(\mathbf R) as those groups whose elements have at most one fixed point.

Let K \mathbf {K} be a field of characteristic 0 0 and let ( G n ( x ) ) n = 0 ∞ (G_{n}(x))_{n=0}^{\infty } be a linear recurring sequence of degree d d in K [ x ] \mathbf {K}[x] defined by the initial terms G 0 , … , G d − 1 ∈ K [ x ] G_0,\ldots ,G_{d-1}\in \mathbf {K}[x] and by the difference equation \[ G n + d ( x ) = A d − 1 ( x ) G n + d − 1 ( x ) + ⋯ + A 0 ( x ) G n ( x ) , for n ≥ 0 , G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots +A_0(x)G_{n}(x), \quad \mbox {for} \,\, n\geq 0, \] with A 0 , … , A d − 1 ∈ K [ x ] A_0,\ldots ,A_{d-1}\in \mathbf {K}[x] . Finally, let P ( x ) P(x) be an element of K [ x ] \mathbf {K}[x] . In this paper we are giving fairly general conditions depending only on G 0 , … , G d − 1 , G_0,\ldots ,G_{d-1}, on P P , and on A 0 , … , A d − 1 A_0,\ldots ,A_{d-1} under which the Diophantine equation \[ G n ( x ) = G m ( P ( x ) ) G_{n}(x)=G_{m}(P(x)) \] has only finitely many solutions ( n , m ) ∈ Z 2 , n , m ≥ 0 (n,m)\in \mathbb {Z}^{2},n,m\geq 0 . Moreover, we are giving an upper bound for the number of solutions, which depends only on d d . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F ( x ) = f ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) F(x)=f(\langle \alpha _{1} , x\rangle , \dots , \langle \alpha _{n} , x\rangle ) where ⟨ α , x ⟩ \langle {\alpha ,x}\rangle denotes the Paley-Wiener-Zygmund stochastic integral ∫ 0 T α ( t ) d x ( t ) \int _{0}^{T} \alpha (t) d x(t) .

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed., Comment: To appear in Trans. Am. Math. Soc

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near ℓ 1 \ell ^1 . For our negative results we generalize an argument of Zienkiewicz.

Let F ^ \hat {F} be a free pro- p p non-abelian group, and let Δ \Delta be a commutative Noetherian complete local ring with a maximal ideal I I such that c h a r ( Δ / I ) = p > 0 \mathrm {char}(\Delta /I)=p>0 . Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64–69] showed that when p ≠ 2 p\neq 2 , the pro- p p congruence subgroup \[ G L 2 1 ( Δ ) = ker ⁡ ( G L 2 ( Δ ) ⟶ Δ → Δ / I G L 2 ( Δ / I ) ) GL_{2}^{1}(\Delta )=\ker (GL_{2}(\Delta )\overset {\Delta \to \Delta /I}{\longrightarrow }GL_{2}(\Delta /I)) \] admits a pro- p p identity, i.e., there exists an element 1 ≠ w ∈ F ^ 1\neq w\in \hat {F} that vanishes under any continuous homomorphism F ^ → G L 2 1 ( Δ ) \hat {F}\to GL_{2}^{1}(\Delta ) . In this paper we investigate the case p = 2 p=2 . The main result is that when c h a r ( Δ ) = 2 \mathrm {char}(\Delta )=2 , the pro- 2 2 group G L 2 1 ( Δ ) GL_{2}^{1}(\Delta ) admits a pro- 2 2 identity. This result was obtained by the use of trace identities that originate in PI-theory.

Let $X$ be a K\"ahler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $\omega$ be a fixed K\"ahler form on $X$. By Yau's theorem, there exists a unique Ricci-flat K\"ahler form $\rho\vert_{X_y}$ for each fiber, which is cohomologous to $\omega\vert_{X_y}$. This family of Ricci-flat K\"ahler forms $\rho\vert_{X_y}$ induces a smooth $(1,1)$-form $\rho$ on $X$ with a normalization condition. In this paper, we prove that the direct image of $\rho^{n+1}$ is positive on the base $Y$. We also discuss several byproducts, among them the local triviality of families of Calabi-Yau manifolds., Comment: 25 pages. Minor changes

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.

Finsler metrics on an open subset in R n with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of F F -rationality under flat base change.

This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k, there exist infinitely many k-rational points on the projective line such that the fiber of f over P is irreducible over k. In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as Part I. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the initial time is -∞, We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.

Let K[?] denote the group algebra of an infinite locally finite group?. In recent years, the lattice of ideals of K[?] has been extensively studied under the assumption that? is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let B be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If B acts nontrivially on V by way of the homomorphism Φ: G → GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal ωK[V] is the unique proper G-stable ideal of K[V] when char K ¬= p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups G of D*, and the action of G on the group algebra K[A], where A is the additive group D + . In particular, properties of the quasi-simple group G come into play only in the final section of this paper.

The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampin's general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.

Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of SO 2n+1 to GL 2n . Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of GLn i (more precisely, cuspidal representations of GL2n i such that the exterior square L-functions have a pole at s = 1). One purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component. In a separate paper, we prove it without any condition using a result on spherical unitary dual due to Barbasch and Moy. We give several applications of the functorial lift: First, we parametrize square integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadic. Second, we give a criterion for cuspidal reducibility of supercuspidal representations of GL m x SO 2n+1 . Third, we obtain a functorial lift from generic cuspidal representations of SO 5 to automorphic representations of GL 5 , corresponding to the L-group homomorphism Sp 4 (C) → GL 5 (C), given by the second fundamental weight.

Let G be a group and let R be a commutative ring. Let T rP icR(RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HDR(G) of T rP icR(RG) consisting of self-equivalences xing the trivial RG-module acts on the cohomology ring H (G; R). The action is functorial with respect to R. The self-equivalences which are 'splendid' in a sense dened by J. Rickard act natural with respect to transfer and restriction to centralizers of p-subgroups in case R is a eld of characteristic p. In the present paper we prove that this action of self-equivalences on H (G; R) commutes with the action of the Steenrod algebra and study the behaviour of the action of splendid self-equivalences with respect to Lannes' T -functor. Introduction In an earlier joint paper (9) with RaphaRouquier I dened the group T rP icR(A) of standard self-equivalences of a derived module category D b (A) for an R-algebra A which is projective as an R-module. For any A-module M let HDM(A) be the subgroup of T rP icR(A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper (11) I showed that, under some hypothesis on M, the group HDM(A) acts in a natural way on the Ext-algebra Ext A (M; M). In case of A being a group algebra RG, with R being a eld of characteristic p and G being a nite group, then J. Rickard denes in (8) what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p-permutation modules induced from diagonal p-subgroups, and by some invertibility condition in the homotopy category. These splendid equivalences induce self-equivalences of the derived categories of centralizers of p-subgroups by the Brauer construction. In (12) I showed that then, for M = R being the trivial module, the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p-subgroups. In the present paper we enlarge these properties still further. The action of self-equivalences of the bounded derived category D b (RG) on H (G; R) commutes with the action of the Steenrod algebra on H (G; R) for any prime p. As consequence of the above statements, any cohomology ring H (G; Fp) denes a functor H (G; Fp) p HD p (B0( p G)) from the modules over the group of derived self-equivalences of the principal block of the group ring Fp HD p (FpG) xing the trivial module to the category of unstable modules Up and similarly in the opposite direction. Obviously, we may restrict to splendid self-equivalences. We shall describe the composition of Lannes' T -functor with the rst functor and the im- age of free unstable modules by the second. Moreover, by a result of Lannes TV (H (G; Fp)) decomposes into direct product of cohomology rings as unstable modules. We shall prove that this decomposition is also a decomposition as modules over the action of splendid self- equivalences. This will give evidence that splendid equivalences are the correct objects to study in this context. The paper is organized as follows. Section 1 recalls the necessary denitions and properties of Even's norm map as it is used here and the denition of the Steenrod operation. In Section 2 it is shown that the normalized part of the outer automorphism group of the

This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing e of a faceted 3-ball P yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of twisted face-pairing shows how to correct this defect of the quotient pseudomanifold P/∈ systematically. The method describes how to modify P by edge subdivision and how to modify any orientation-reversing face-pairing ∈ of P by twisting, so as to yield an infinite parametrized family of face-pairings (Q, δ) whose quotient complexes Q/δ are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings ∈, it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex Q/δ of a twisted face-pairing is a closed 3-manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to Q. (2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct? We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings. We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds Q/δ and of the original pseudomanifold P/∈ (with vertices removed). We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurston's eight 3-dimensional geometries.

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type(S+)L(S_{+})_{L}acting from a real, reflexive and separable Banach spaceXXintoX∗.X^{*}.This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given inlp, p>2,l^{p},~p>2,illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type(S+)).(S_{+})).Applications to nonlinear Dirichlet problems have appeared elsewhere.

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp’s conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

An S L n SL_n -character of a group G G is the trace of an S L n SL_n -representation of G . G. We show that all algebraic relations between S L n SL_n -characters of G G can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space X , X, with π 1 ( X ) = G . \pi _1(X)=G. We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of S L n SL_n -representations of groups. The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of M M which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the S L 2 SL_2 -character variety of π 1 ( M ) . \pi _1(M). This paper provides a generalization of this result to all S L n SL_n -character varieties.

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories. This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type A A and B B , giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.

This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pullbacks of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition k(P) and k(f(P)) differ when P is an algebraic point on a curve C and f : C --C' is a nonconstant morphism of curves. In particular, it is demonstrated that k(P) 74 k(f(P)), with at most finitely many exceptions, whenever the degrees of P and f are sufficiently small, relative to the difference between the genera g(C) and g(C'). The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2. Let C be a curve defined over a number field k, and let X be a regular model for C over the ring of integers R of k. In [V 3], Vojta proves that, for any c > 0, all points P E C(k) of bounded degree satisfy the inequality (0.0.1) hK(P) < da(P) + EhA(P) + 0(1), where A is any ample divisor on C, hA and hK are Weil heights associated with A and the canonical divisor K, respectively, and da(P) is the arithmetic discriminant of P. The arithmetic discriminant depends on the choice of a regular model X; it is defined as (Hp.(Wx/B+HP)) [k(P) : k] where Hp is the arithmetic curve on X corresponding to P and Wx/B is the sheaf of relative differentials of X over B = Spec R. The difference between arithmetic discriminants derived from different regular models is always bounded. Hence, (0.0.1) is not affected by one's choice of a regular model. The arithmetic discriminant is related to d(P), the normalized field discriminant of the field of definition of P, which is defined as d(P) '_ log IN|Dk(p)/k [k(P): ] One may obtain da(P) by adding to d(P) terms corresponding to singularities of Hp at finite places and v-adic distances between the conjugates of P at infinite places v (see [V 2, Section 3]). The precise nature of the relationship between Received by the editors November 30, 1999 and, in revised form, February 25, 2000. 2000 Mathematics Subject Classification. Primary 11G30, 11J25. ()2001 American Mathematical Society

In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category., 18 pages, 6 figures, to appear in Trans. Amer. Math. Soc

In this paper, we study some algebraic topology aspects of String c ^c structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of S p i n c ( n ) Spin^c(n) . We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1–40] to correspond to String c ^c structures of various levels and give vanishing results for them.

For the open unit disc $\mathbb{D}$ in the complex plane, it is well known that if $\phi \in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetilde{\phi}$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform., Comment: Fixed few typos. Minor changes. To appear in Trans. Amer. Math. Soc

The distance from the origin in the word metric for generalizations F (p) of Thompson’s group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F (p). This interpretation of the metric is used to prove that every F (p) admits a quasiisometric embedding into every F (q), and also to study the behavior of the shift maps under these embeddings. The first example of a finitely presented, infinite, simple group was discovered by R.J. Thompson in 1965. This group G arose in the study of logic and it has many interesting descriptions. In 1974, Higman [10] used a description as the group of automorphisms of the Jonsson-Tarski algebra to generalize G to an infinite family of groups. G has another description as a group of homeomorphisms of the Cantor set. It has subgroups which may be described as homeomorphism groups of different objects, for instance, the circle and the unit interval. Among the people studying these generalizations are Brown [3] and Bieri and Strebel, in a set of unpublished notes [1]. In this paper we will concentrate on the groups F (p), which correspond to the groups Fp,∞ in [3]. These families of groups have also been extensively studied by Brin and Guzman [2]. The group F (2), commonly known as Thompson’s group F , can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the interval [0, 1] which have breakpoints only at dyadic points and whose slopes, where defined, are powers of two. In 1984 Brown and Geoghegan [4] found it to be the first example of a finitely presented, infinite-dimensional, torsion-free FP∞ group. This fact has been extended to all F (p) in [3], and studied by Stein [11], where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered. Cleary [7] has studied these properties for groups of piecewise-linear homeomorphisms with irrational breakpoints and slopes. In [5] Burillo established the relationship between the word metric of Thompson’s group F and an estimate of the distance derived from the unique normal form of the elements. This algebraic estimate is quasi-isometric to the word metric and was used to prove that some subgroups are nondistorted in F . In this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams (see [6]), show that this is quasi-isometric to the word metric, and use this interpretation to prove that some embeddings of F (p) into F (q), including some of those outlined in [2], are quasi-isometric. The interpretations of these embeddings in terms of Received by the editors September 25, 1998 and, in revised form, August 11, 1999. 2000 Mathematics Subject Classification. Primary 20F65; Secondary 20F05, 20F38, 20E99, 05C25. c ©2000 American Mathematical Society

In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e. The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in e. 1. THE MODEL AND MAIN RESULT In this paper we present a general technique for establishing analyticity of solutions of systems of partial differential equations which depend analytically on a parameter e. The method works not only for boundary value problems but also for free boundary problems. In this latter context it can be used to establish long time existence of transient solutions, and also to study the existence of spatially asymmetric steady solutions. Since free boundary problems are typically more challenging than their boundary-value counterparts, we shall concentrate here on a free boundary problem from developmental biology, namely, a model of tumor growth. To further exemplify the generality of our approach an instance of a boundary value problem (in a fixed domain) is presented in the last section of the paper. A variety of other problems are amenable to the same analysis, including, in particular, the Hele-Shaw model of fluid flow [11]. Within the last several decades a number of mathematical models have been developed that aimed at describing the evolution of carcinomas (see. e.g., [1, 5, 6, 8, 12, 13] and the references cited there). The main objective of these models has been to qualitatively describe, under various simplifying assumptions, the growth and stability of tumor tissue. Analysis and simulations of such models are helping to assess the relative importance of various mechanisms affecting tumor growth as well as the efficacy of certain cancer treatments. On the other hand, the description of the stationary (dormant) configurations that arise from the models has only been addressed in the case of spherical tumors, but otherwise it remains largely unexplored. In this paper we develop a method for establishing analyticity of Received by the editors August 17, 1999. 1991 Mathematics Subject Classification. Primary 35B32, 35R35; Secondary 35B30, 35B60, 35C10, 35J85, 35Q80, 92C15, 95C15.

We study a class of endomomorphism algebras of certain q-permutation modules over the Hecke algebra of type B, whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed. The notion of a q-Schur algebra was introduced by Dipper and James [DJ2], who used these algebras to parametrize the irreducible representations of the finite general linear groups in non-describing characteristics. With hindsight, these algebras had already appeared earlier in an entirely different quantum group context [Ji] inspired by physics. In [Ji] Jimbo considered the endomorphism algebras of tensor spaces as Hecke algebra modules. In his context, a q-Schur algebra can be viewed as a quotient of the quantized enveloping algebra associated to gln. In [PW], these algebras were shown to be quasi-hereditary. The quasi-heredity property is an embodiment in classical algebra of the geometric derived category stratification exhibited by perverse sheaves [PS]. It means more applications can be deduced from a ring-theoretic point of view (see e.g., [DPS3]), and the possibility is raised of even deeper results in the future, as suggested by [CPS2]. Certainly, these algebras play a central role in the representation theories of the finite and quantum general linear groups. Naturally, one asks: Are there such algebras for types other than A? Our paper [DS] showed that there were similar quasi-hereditary quotients of quantized enveloping algebras for all types of root systems. However, no connection with Hecke algebras and finite groups of Lie type was found there. This paper aims at the same question and constructs possible algebras directly from Hecke algebras (hence from finite groups of Lie type). We restrict attention to the type B case. Imitating the definition of a q-Schur algebra, we introduce the notion of a q-Schur algebra. These algebras are the endomorphism algebras of certain modules over the Hecke algebra of type B — called “tensor” spaces — whose summands involve not only parabolic subgroups but also quasi-parabolic subgroups of the Weyl group of type B. A main result of our paper shows that qSchur algebras are quasi-hereditary. We speculate that similar constructions exist for other classical types (only type D remains, actually, since type C is equivalent Received by the editors March 3, 1997 and, in revised form, October 28, 1998. 2000 Mathematics Subject Classification. Primary 20C08, 20G05, 20C33. The authors would like to thank ARC for support under the Large Grant A69530243 as well as NSF, and the Universities of Virginia and New South Wales for their cooperation. The first author also thanks the Newton Institute at Cambridge for its hospitality. c ©2000 American Mathematical Society

Let k be a finite field, k Ik the degree n extension of k, and G = GLn(k) the general linear group with entries in k. This paper studies the "generalized Steinberg" (GS) representationis of G and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of G is in natural one-one correspondence with the set of Galois orbits of characters of kx, the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are "generic" (have Whittaker vectors). Let k = Fq be a finite field of cardinality q, let kIkm Ik be,_ respectively, an algebraic closure of k and the degree rm extension of k contained in k. Let q x _ Xq denote the Frobenius automorphism of klk and of any subextension km1k. Let G = GL (k) be the group of non-singular n x n matrices with entries in k. Also write Gm = GLrn(k) for any m > 1. In a signally important paper published in 1955 [GR] J. A. Green showed how to calculate formulas for the irreducible characters of G. In Green's work appeared for the first time general character formulas for the cuspidal representations and for a family of "generalized Steinberg" (GS) representations. For their study of the "level zero" discrete series characters of unit groups of simple algebras over a p-adic field the authors need diverse characterizations of cuspidal and GS characters of finite general linear groups and to be able to pass between these different characterizations. In this paper we give these characterizar tions and prove their equivalence. It is fruitful to view the set of GS representatio:ns as a class of representations of G which contains the class of cuspidal representations as a subelass. Of course the cusp form property, the propert;y of not being a component of Ind u 1 for any unipotent radical U :& (I) of a parabolic subgroup of G, clearly distinguishes the class of cuspidal representations from all other representations of G. However, other important properties which are usually associated to the class of cuspidal representations generalize to the union of the two classes of representations of C. Received by the editors May 26, 1997 and, in revised form, April 18, 1998 and June 26, 1998. 1991 Mathematics Subject Classification. Primary 22E50, 11T24.

We consider the problem of counting solutions to a trinomial Thue equation — that is, an equation ( ∗ ) | F ( x , y ) | = 1 , \begin{equation*} |F(x,y)| = 1,\tag {$*$} \end{equation*} where F F is an irreducible form in Z [ x , y ] Z[x,y] with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the “Thue-Siegel principle" and its relation to ( ∗ ) (*) . In this paper we give specific numerical bounds for the number of solutions to ( ∗ ) (*) by a somewhat different approach, the difference lying in the initial step — solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.

Let FFn be a family of subgroups of EnX which is closed under taking subgroups and conjugates. Such a family has a classifying space, BYFn, and we showed in an earlier paper that a cormpatible choice of TFn for each n gives a simplicial monoid JJn BTF,, which group completes to an infinite loop space. In this paper we define a filtration of the associated spectrum whose filtration quotients, given an extra condition on the families, can be identified in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The first is related to elementary abelian p-groups, and the second gives a new expression for the desuspension of Sp""' (SO)/Spm--I (SO) as a suspension spectrum.

Let LR(H) denote the set of all closed linear relations on a Hilbert space H (which contains all closed linear operators on H). In this paper, for every E E LR(H) we define and study two associated linear operators on H, cos(E) and sin(E), which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of E, a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the cos(E) and sin(E) operators to operator theory will be given in a forthcoming paper.

This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map f into ag raph of a multi-valued representable map F. The multi-valued representable maps have well developed continuity properties and admit a nite coding that per- mits treating them by combinatorial methods. We provide the construction of the homomorphism F induced by F such that F = f. The presented construction does not require subsequent barycentric subdivisions and simpli- cial approximations of f. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.

We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory “ZF+DC+ω1 is the only regular, uncountable cardinal ≤ ωω1+1.” This gives a partial answer to a question of Woodin. In 1994, J. Steel proved what had long been suspected: that assuming AD, the regular cardinals of L(R) below Θ are all measurable, where Θ is the least ordinal onto which the real numbers R cannot be mapped [St3]. This remarkable theorem motivated the work of [A], which showed essentially that the cofinality of any regular cardinal below Θ can be changed to ω without perturbing cardinal structure. Specifically, the following theorem was proved. Theorem 0.1. Assume V′ AD, and let V = L(R)V . Then for any subset A of the regular, uncountable cardinals below Θ, there is a partial ordering P ∈ V and a symmetric inner model N of ZF such that (1) V ⊆ N ⊆ VP , (2) N and V contain the same cardinals, (3) Θ = Θ, (4) for all κ, cof(κ) = ω if cof(κ) ∈ A and cof(κ) = cof(κ) otherwise, and (5) the measurable cardinals not in A remain measurable in N. What was left unanswered is whether similar facts can be proven if the cofinality of κ < Θ is to be made uncountable. Consideration of this question was the genesis of this paper. The principal tool, infinite–exponent partition sequences, proved as interesting as its application. We begin, in the first three sections, with background material on infinite– exponent partition properties of cardinals, their abundance in models of AD, and their use in “Magidor–like” forcing to change cofinalities. The second three sections deal with infinite–exponent partition properties of cardinal sequences. With appropriate notation, the basic theory of partition sequences and the associated Magidor–like forcing exactly parallels the cardinal theory. In “Finite Support,” we apply the theory to show the extent to which possibly uncountable cofinal sequences can be added if we sacrifice DC. In “Countable Support,” we preserve DC. Received by the editors September 30, 1997. 1991 Mathematics Subject Classification. Primary 03E15, 03E35, 03E60. The first author’s research was partially supported by PSC–CUNY grant 665337. c ©1999 American Mathematical Society

We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform

We define a multiplication on the cyclic homology of a commutative, cocommutative bialgebra H with “superproduct.” In the case when H is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator. (The Loday-Quillen multiplication is trivial in that case.) In this paper we define an associative multiplication on the cyclic homology of certain commutative algebras H . The definition is based on the Cartesian product of sets instead of the shuffle product as in the case of [LQ] and [Ga]. When H contains Q our product is compatible with the Hodge decomposition of HC∗(H) so it has the form: HC n (H)⊗HC(j) m (H) → HC n+m (H). The existence of this multiplication in the case H = K is classical since it is just given by the tensor product of complex line bundle. However its significance came to our attention in joint work with John Klein as a consequence of our study of fiberwise Morse theory on circle bundles [IK] and in other joint works with R. Penner and R. Hain. By doing Morse theory on circles John Klein and I realized that there was a rationally nontrivial associative multiplication on the cyclic homology of the integral group ring on any cyclic group and that it had important consequences for pseudoisotopy theory. This will be explained in a later paper. In this paper we show that this multiplication in cyclic homotopy is defined first in the case when the algebra is H = KM = K[M ] where M is the underlying additive group of a ring, then in the case when H is a Hopf algebra with “superproduct” in the sense of Gaucher [Ga] and finally in the more general case of any “μ-algebra” (Example 12.4). In fact we do not need additive inverses in M nor do we need an antipode for H so we can take M to be a monoid and H to be a bialgebra. When H = K, i.e. M = 1, the cyclic homology becomes a divided power algebra which is of course isomorphic to a polynomial algebra in one generator if K is a field of characteristic zero. We assume throughout the paper that K is a commutative ring with 1 and that all tensor products are taken over K. This includes those tensor products which appear in the definition of cyclic homology. We also use Connes’ original definition that a cyclic object in a category C is a covariant functor Λ → C where Λ is Connes’ cyclic category. Since Λ ≈ Λ this is the same as a contravariant functor Λ → C. An outline of the paper is as follows. In the first section we give a topological definition of the multiplication in HC∗(KM). The construction is so elementary that Received by the editors April 14, 1994. 1991 Mathematics Subject Classification. Primary 18G60; Secondary 16W30. This research was supported by NSF grant no. DMS 90 02512. c ©1999 American Mathematical Society

We give a characterization theorem for the E(A)-normalized subgroups of G2 (A), where A is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking. INTRODUCTION In this paper we prove a theorem (3.6) which characterizes the E(A)-normalized subgroups of G2(A) for an arbitrary commutative ring A. The theorem brings to completion the larger project of finding characterization theorems for E(A)normalized subgroups of simple Chevalley-Demazure group-schemes G(A) of rank > 2 over arbitrary commutative rings. The bulk of t;he project was done by Vaserstein in his essential paper [16]. There he proved the standard characterization theorem, given by a "ladder" corresponding to an ideal, for all commutative rings for all G of rank > 2 except those of types B2= C2 and G2. Vaserstein also suggested in that paper that previous work by Abe [1] and Abe-Suzuki [3] could be improved to yield more. This was accomplished by Abe in [2], wherein he showed that E(A)normalized subgroups of G(A) could be characterized by ladders determined by "admissible pairs" of ideals, with the exception that for G of type B2 = C2 or G2 rings A having residue class fields of two elements must be excluded. Our paper [8] subsequently succeeded in removing this restriction for G of type C2 (and so also for G of type B2) by recognizing that the presence of residue fields of two elements required lowering the bottom rung of the ladder. This left only the case G of type G2 to be finished; the present paper does so by applying the "lower bottom rung" idea of [8] (see also [7]) to this case. We now give a more explicit description of our results. Let A be an arbitrary commutative ring and denote by G2(A) the group of type G2 defined over A. The main theorem (3.6) is that a subgroup N of G2(A) is E(A)-normalized if and only if C(J, J') C N C G(J, J') for a pair of ideals J, J' satisfying the condition J3 C J' C J. Here, J3 denotes the ideal of A generated by all elements of the form x3 and 3x for all x in J, so that (J, J') is an admissible pair in the sense of Abe. The bracketing groups C(J, J') and G(J, J') are defined in Section 1 below. In essence, C(J, J') is the mixed commutator group [E(A), E(J, J')], where E(J, J') is the E(A)-normalized subgroup generated by all elements x,(j) with a long and Received by the editors April 1, 1997 and, in revised form, May 22, 1997. 1991 Mathematics Sutbject Classification. Primary 20H05; Secondary 20G35. Research partially supported by NSA grant MDA 904-94-H-2008 and NSF grant DMS-9622899. (?)1999 American Mathematical Society

We establish an LP a priori estimate for the Tricomi equation. Our main tool is Mihlin's multiplier theorem combined with well-known estimates of the Newtonian potential. 0. INTRODUCTION The purpose of this paper is to establish an LP a priori estimate for the Tricomi equation in the upper half-space. The Tricomi equation arises in transonic gas dynamics, and is a typical model equation of changing type. It has been extensively investigated from the various viewpoints. The Tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary, which is our viewpoint in this paper. We can formulate the Dirichlet boundary value problem in the upper half-space as follows. 02 + Yacuz=f, in R+ (0.2) u(0,x) = (x), on MT, where R+ {(y, x) : y > 0, x E Rn-} and /x is the Laplacian in the x variable. An L2 a priori estimate can be obtained very easily. Following the presentation in [8], we suppose that u E Co (Rn), and multiply the equation (0.1) by &92U Then, we integrate over R,r to obtain (0.3) IR ( dy2x)2Jj+ V dydx Rn-1 o 9Y2 Rn-1 o a'y 2IRn1l |Vq$ dx + j-1 f ,3 dydx, where Vx stands for the gradient in x E R n-. This yields (0.4) 2U + \F 1 +YX 4Y21 L2(Rn) Dy L2(Rn) L2(Rn) < M(||f 1|L2(Rn) + 11VX01iL2(Rn-1))7 Received by the editors December 30, 1996 and, in revised form, February 10, 1998. 1991 Mathematics Subject Classification. Primary 35J70, 35B45.

LetGGbe a finitely generated residually finite group and letan(G)a_n(G)denote the number of indexnnsubgroups ofGG. Ifan(G)≤nαa_n(G) \le n^{\alpha }for someα\alphaand for allnn, thenGGis said to have polynomial subgroup growth (PSG, for short). The degree ofGGis then defined bydeg⁡(G)=lim suplog⁡an(G)log⁡n\operatorname {deg}(G) = \limsup {{\log a_n(G)} \over {\log n}}. Very little seems to be known about the relation betweendeg⁡(G)\operatorname {deg}(G)and the algebraic structure ofGG. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that ifH≤GH \le Gis a finite index subgroup, thendeg⁡(G)≤deg⁡(H)+1\operatorname {deg}(G) \le \operatorname {deg}(H)+1. A large part of the paper is devoted to the structure of groups of small degree. We show thatan(G)a_n(G)is bounded above by a linear function ofnnif and only ifGGis virtually cyclic. We then determine all groups of degree less than3/23/2, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval(1,3/2)(1, 3/2). Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.

We study geometric properties of the metric projection r: M S of an open manifold M with nonnegative sectional curvature onto a soul S. ir is shown to be C? up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of S also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of M, and stu(dy the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of M. The resolution of the Soul conjecture of Cheeger and Gromoll by Perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected. Orne of the key results in [14] is that the metric projection wF: M -> S which maps a point p in M to the point 7r(p) in S that is closest to p is a Riemannian submersioni. Perelman also observed that this map is at least of class C1, and later it was shown in [9] that 7w is at least C2, and C? at almnost every point. The existence of such a Riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of M inust be special. The purpose of this paper is to illustrate this in several different directions. In fact, we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map -w. The paper is essentially structured as follows: After introducing notation and recalling some basic results in sectioni 1, we establish in section 2 the existence of maps similar to -w for any submanifold of M homologous and isometric to the soul, and use this to prove the existence of convex tubular neighborhoods for them. In particular, this also proves: Theorem 2.5. The unit normal bundle vti(S) admits a metric with nonnegative sectional curvature. As a consequence, open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound (see also [81). Theorem 2.5 also implies that nontrivial plane bundles over a Bieberbach maanifold do not admit nonnegatively curved metrics, thereby providing a shorter proof of the main result in [13]. We are grateful to the referee for pointing out this fact to us. Properties such as these are established via stanidard submersion techniques, but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of M far away from the soul. One way around Received by the editors August 18, 1997. 1991 Mathematics Subject Classification. Primary 53C20.

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group. It was recently proved in [1] that finite systems of equations over the Q–completion of a finitely generated free group are algorithmically solvable. In this paper we generalize the results of [1] to the case of the Q–completion G of an arbitrary torsion–free hyperbolic group G with generators d1, . . . , dN . (A detailed definition of the Q–completion of a group can be found in [1].) Main Theorem. Let G be a torsion–free hyperbolic group. Then there exists an algorithm that decides if a given finite system of equations over the Q–completion of G has a solution, and if it does, finds a solution. By a triangular equation we mean an equation with at most three terms. Given a system S of equations in G, we may assume that all equations in S are triangular. Indeed, an arbitrary equation x1 1 x e2 2 . . . x en n = 1 (where xi stands either for a variable or for a constant, ei = ±1) is equivalent to a finite system of triangular equations: x1 1 x e2 2 y −1 1 = 1, y1x e3 3 y −1 2 = 1, . . . , yn−2x en−1 n−1 x en n = 1. Adding a finite number of new variables and equations, we may also assume that S contains only equations with coefficients in G. To achieve this, we replace every constant of the form d m n by a new variable z satisfying the equation z = d. From now on, we fix a finite system S of triangular equations over G with coefficients in G. We will reduce the system S to a finite set of systems in a specific hyperbolic group G ∗ 〈t1, . . . , tψ(m)〉, where the number ψ(m) can be determined effectively given the number m of equations in the original system S. The resulting systems are accompanied by the restriction that some of the variables belong to a certain subgroup G ∗ 〈t1, . . . , ti〉 of G ∗ 〈t1, . . . , tψ(m)〉, i.e. do not contain certain t’s. To each of the systems we can apply the (slightly modified) method of Rips and Sela [4] to see if they are decidable. If none of them is consistent, then the original system S has no solution; if at least one has a solution, then it is possible to find a corresponding solution to the system S. Suppose that the system has a solution in G; let {X1, . . . , XL} be a solution with the minimal possible number of roots. Since the solution belongs to G, it is contained in a certain group K, obtained from G by adding finitely many roots. It is the union of a chain of subgroups Hi defined as follows. Let G = H0. Received by the editors August 14, 1996. 1991 Mathematics Subject Classification. Primary 20E05, 20F10. The first author was supported by grants from NSERC and FCAR. The third author was supported by the NSF grant DMS-9103098. c ©1999 American Mathematical Society 2961 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2962 O. KHARLAMPOVICH, E. LIOUTIKOVA, AND A. MYASNIKOV Step 1. Consider pairwise nonconjugated cyclically minimal primitive elements u1, . . . , uk1 ∈ G, |u1| ≤ . . . ≤ |uk1 | (here |u| denote the length of u in G), and add roots t1, . . . , tk1 , such that uj = t sj j . (Notice that ui+1 does not become a proper power after we add roots t1, . . . , ti.) The corresponding groups are denoted by H1, . . . , Hk1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step 2. Consider pairwise nonconjugated primitive elements uk1+1, . . . , uk2 ∈ H1, cyclically reduced in the amalgamated product, each having the reduced form u = t1 1 c1 . . . t αk 1 ck, where αi 6= 0, αi ∈ Z, ci ∈ G, |uk1+1|H1 ≤ . . . ≤ |uk2 |H1 ; and add roots tk1+1, . . . , tk2 , such that uj = t sj j , to the group Hk1 . The corresponding groups are denoted by Hk1+1, . . . , Hk2 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step i+1. Suppose that H1, . . . , Hki have been constructed. Consider pairwise nonconjugated primitive elements uki+1, . . . , uki+1 ∈ Hi, cyclically reduced in the amalgamated product, each having the reduced form t1 i c1 . . . tk i ck, where αi 6= 0, αi ∈ Z, ci ∈ Hi−1 ( ck is not a power of ui, because the elements are cyclically reduced), |uki+1|Hi ≤ . . . ≤ |uki+1 |Hi , and add roots tki+1, . . . , tki+1 , such that uj = t sj j , to the group Hki . The corresponding groups are denoted by Hki+1, . . . , Hki+1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Finally, for some number i one has K = Hki+1 . The group Hki+1 is called the group at level i, corresponding to the sequence u1, . . . , uki+1 . The group Hi will be called the group of rank i. We also order the set of tj ’s: tk < tl if k < l. Let Fi be the free group with the same generating set as Hi, i = 1, . . . , k1, and let β : Hi → Fi be a section, i.e. a mapping of sets such that π ◦ β = id. For our purposes, β has to be of specific form; we will define it explicitly in section 2. For X ∈ Hi we will call β(X) the canonical representative of X . Definition 1. By an equational diagram over a group Hi we mean an equation X1 . . . Xn = 1 together with a solution A1, . . . , An and a diagram over Hi having β(A1) . . . β(An) as its boundary label. An equational triangle is an equational diagram corresponding to a triangular equation. A system of equational diagrams is a system of equations together with the system of diagrams such that the solution associated to each equational diagram is a solution of the whole system. A free equational diagram is an equational diagram with no cells; the corresponding equations are called free equations. If the level i of the group Hki+1 is greater than zero, we can use the method of [1] to reduce each equation of the system S to a bounded number of free equations and at most one equation over the group in the previous rank. We can proceed in the same way until we reach level 0 (i.e. the group Hk1), for which we need to develop a separate method. In what follows we will denote Hk1 by H . 1. Some properties of the Cayley graph of H We will frequently use the following lemma: Lemma 1. ([3, Lemma 1.5]) For each geodesic triangle [x1, x2, x3] in a δ-space, there are points yi ∈ [xi−1, xi+1] (indices are considered modulo 3) such that d(xi, yi−1) = d(xi, yi+1) = (xi−1 · xi+1)xi , License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE Q–COMPLETION OF A TORSION–FREE HYPERBOLIC GROUP 2963

The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic p of the ground field is greater than 2). For the symmetric group there are exactly p 1 such ideals and for the alternating group there are (p 1)/2 of them. The description is obtained in terms of the annihilators of certain systems of the 'completely splittable' irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the 'Mullineux conjecture' proved recently by FordKleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.

The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space. 1. PRELIMINARIES AND STATEMENT OF RESULTS An isometric immersion f: S' -* SV, m > 2, of the Euclidean mr-sphere Sk of constant curvature k into the unit sphere Sv of a Euclidean vector space V is said to be a spherical minimal immersion if f is minimal. By a result of Takahashi [11], f exists if k = m/Ap for some p > 1, where AP = p(p+m1) is the p-th eigenvalue of the spherical Laplacian As7m on Sm := S', and, in this case, each component O f q e V*, of f is a spherical harmonic of order p on Sm; an eigenfunction of Asm with eigenvalue AP. We scale the metric on the domain to curvature one and call f Sm S Sv a spherical minimal immersion of (algebraic) degree p. Because of the scaling, f is a homothetic (minimal) immersion with homothety Ap/m: (1) (f* (X) X f* (Y)) = Ap/m(XI Y), where X, Y are vector fields on Sm. Let HP denote the Euclidean vector space of spherical harmonics of order p on Sm endowed with the L2-scalar product (suitably scaled). The universal example of a spherical minimal immersion is the standard minimal immersion fm,p: Sm SHP whose components (relative to an orthonormal basis in HP ) are orthonormal. For p = 2, fm,2: Sm S-2 is the classical Veronese map. Unless relevant, we suppress m from the notation and write the standard minimal immersion as fp: Sm SEP. A fundamental problem raised by M.DoCarmo and N.Wallach [3] is to find lower bounds for the range dimension of spherical minimal immersions. The main result of this paper, Theorem 4, solves this problem by giving lower bounds in terms of the dimension of the domain, the degree, and differential geometric properties of the immersions such as symmetries and higher order isotropy. The only previously known general lower bound was given by J.D.Moore [10] who proved that, for a spherical minimal immersion f: Sm + Sv of degree > 2, we have dim V > 2m. DoCarmo and Wallach conjectured [3] that the lower bound Received by the editors April 20, 1997. 1991 Mathematics Subject Classification. Primary 53C42. ?)1999 American Mathematical Society

Let S be an F -finite regular local ring and I an ideal contained in S. Let R = S/I. Fedder proved that R is F -pure if and only if (I [p] : I) * m[p]. We have noted a new proof for his criterion, along with showing that (I [q] : I) ⊆ (τ [q] : τ), where τ is the pullback of the test ideal for R. Combining the the F -purity criterion and the above result we see that if R = S/I is F pure then R/τ is also F -pure. In fact, we can form a filtration of R, I ⊆ τ = τ0 ⊆ τ1 ⊆ . . . ⊆ τi ⊆ . . . that stabilizes such that each R/τi is F -pure and its test ideal is τi+1. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let R = T/I, where T is either a polynomial or a power series ring and I = P1 ∩ . . . ∩ Pn is generated by monomials and the R/Pi are regular. Set J = Σ(P1 ∩ . . . ∩ Pi ∩ . . . ∩ Pn). Then J = τ = τpar. This paper concerns the study of the test ideal in F -finite quotients of regular local rings. Test elements play a key role in tight closure theory. Once known, they make computing tight closures of ideals and modules easier. In fact, in an excellent Gorenstein local ring with an isolated singularity, R/τ ∼= Hom(I∗/I, E) where I is generated by a system of parameters that are test elements and E is the injective hull of R (see [Hu1] and [S1]). We also know for parameter ideals I that I : τ = I∗. Thus knowing τ is basically equivalent to knowing the tight closure of a system of parameters which is contained in the test ideal. A recent paper of Huneke and Smith [HS] links tight closure to Kodaira vanishing for graded rings R with characteristic either 0 or p where p 0. Recall that the a-invariant for a graded ring R, denoted a, is equal to −min{i|[ωR]i 6= 0}, where ωR is the canonical module for R. If R is Gorenstein, then ωR = R(a). Huneke and Smith prove that the test ideal is exactly the ideal generated by elements of degree greater than the a-invariant of R if and only if a strong Kodaira vanishing holds on R. A recent paper of Hara [Ha] confirms that this strong Kodaira vanishing holds in finitely generated algebras over a field of characteristic zero. In this paper we study test ideals of F -finite rings which are reduced quotients of a regular local ring. Reduced quotients of F -finite regular local rings have been studied by both Fedder [Fe] and Glassbrenner [Gl]. Fedder’s work concerns F purity aspects of these rings, and Glassbrenner’s results use Fedder’s techniques to examine strong F -regularity. The object that plays a key role in their work is Received by the editors November 4, 1996. 1991 Mathematics Subject Classification. Primary 13A35.

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. We study the problem of how weak8-closedness of some translation invariant subspaces of B(G) is related to the structure of G. Moreover, we prove that for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak8-continuous only when H is open in G. INTRODUCTION Let G be a locally compact group, and let B(G) be the Fourier-Stieltjes algebra of G as defined by Eymard [8]. Recall that B(G) is the linear span of all continuous positive definite functions on G and can be identified with the Banach space dual of C*(G), the group C*-algebra of G. The space B(G), with the xlorm as dual of C*(G), is a commutative Banach *-algebra with pointwise multiplication and complex conjugation. The Fourier algebra A(G) of G is the closed *-subalgebra of B(G) generated by the functions in B(G) with compact support. In particular, A(G) is contained in Co(G), the algebra of complex valued continuous functions on G vanishing at infinity. As is well known A(G) is weak*-dense in B(G) if and only if G is amenable. In [3] translation invariant *-subalgebras A of B(G) were studied, and it was shown that if such A is weak*-closed and point separating, then it must contain A(G). However, apart from this, very little seems to be known about weak*-closed subspaces of B(G). The first purpose of this paper is to investigate the relation between weak*closedness of certain interesting norm-closed translation invariant subspaces of B(G) and the structure of G. Secondly, we solve the problem of whexl, for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak*-continuous. A brief outline of the paper is as follows. In Section 2 we estsablish for almost connected locally compact groups G the relation between weclk*-closed.ness of Bo(G) = B(G) n Co(G) in E3(G) and the structure of G (Theorem 2.10). The key result is that for a connected Lie group G, Bo(G) is weak*-closed in B(G) if and only if G is a reductive Lie group with compact centre and Kazhdan's property (T). Received by the editors December 1S, 1995. 199l Mathematics Subject Classification. Primary 22D10, 43A30. Work supported by NATO collaborative research grant CRG 940184. (g)1997 American Mathematical Society

In this paper, we study the topology of spaces of n n -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the E 2 E^2 term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163–221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.