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The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus

This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in \mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $f(x)\in\mathbb{Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in $\overline{\mathbb{F}}_p$ for any p not dividing N., Comment: 24 pages, minor corrections

Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain R R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R R , the Grothendieck group G 0 ( R ) ¯ \overline {G_0(R)} of finitely generated R R -modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R R is the cone in G 0 ( R ) ¯ R \overline {G_0(R)}_{\mathbb R} spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ϵ i = 0 , ± 1 \epsilon _i = 0, \pm 1 ( d / 2 > i > d d/2 > i > d ), we shall construct a d d -dimensional Cohen-Macaulay local ring R R (of characteristic p p ) and a maximal primary ideal I I of R R such that the function ℓ R ( R / I [ p n ] ) \ell _R(R/I^{[p^n]}) is a polynomial in p n p^n of degree d d whose coefficient of ( p n ) i (p^n)^i is the product of ϵ i \epsilon _i and a positive rational number for d / 2 > i > d d/2 > i > d . The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold., Paper completely rewritten. Main sutured manifold theory results have been moved to a separate paper

Let k be an infinite perfect field and let X = SpecR, where R is the localization of a k-algebra B of finite type with respect to a regular prime ideal p ∈ SpecB. LetD be a divisor onX with a decompositionD = ∪i=1Di into its irreducible components Di such that each Di is nonsingular and they intersect transversally. Consider Y = X − D, a deleted neighbourhood of the closed point x0 of X . Let E be a vector bundle on Y . In this set up, E is trivial in the following cases : (i) dim(X) ≤ 3 (Gabber [Ga1]) ; (ii) t = 1 (Bhatwadekar-Rao [BR]) ; (iii) rankE ≥ min{[dimR 2 ], t} (Rao [Ra1]). The above exposition is borrowed from a paper of Nisnevich [Ni], which contains some beautiful applications of these ‘purity theorems’. In the same set up, now let y ∈ Y be a closed point and let my be the corresponding maximal ideal of OY,y. It is natural to ask whether my is a complete intersection. Observing that the conormal bundle my/my is trivial, we consider a more general situation : Let I be the defining ideal of a closed subscheme Z of Y such that the conormal bundle I/I2 (defined on Z) is trivial. Is I a complete intersection? The questions we are addressing in this paper are a little stronger in nature : Can a given basis of I/I2 be lifted to a minimal set of generators of I? We remark that with the hypothesis on X , each component Di is defined by a single equation fi which vanishes at the closed point x0 of X . Let m be the maximal ideal at x0. The condition of transversality of intersection of Di’s is equivalent to saying that the images of f1, · · · , ft in the A/m-vector space m/m2 are linearly independent. Keeping this discussion in mind, we now give an excerpt of our main results in commutative algebraic terms (see 4.2, 4.3, 4.4).

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer $n \geq 1$, if the entrywise application of a smooth function $f : (0,\infty) \to \mathbb{R}$ preserves the set of $n \times n$ positive semidefinite matrices with positive entries, then $f$ and its first $n-1$ derivatives are non-negative on $(0,\infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In recent works with Belton-Guillot-Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, real-analytic versions at the origin of the Horn-Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first $n$ nonzero Taylor coefficients at individual points rather than on all of $(0,\infty)$. A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy's (1841) determinant (whose matrix entries are geometric series $1 / (1 - u_j v_k)$), as well as of a determinant of Frobenius [J. reine angew. Math. 1882] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings., Comment: Final version, to appear in Transactions of the American Mathematical Society. 19 pages, no figures

We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in B_n} (\mbox{deg}\, v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)+\sum_{j=0}^{n} (k_j +6)} = \infty. \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998., Comment: 45 pages, 19 figures; to appear in TAMS

In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in Rd. One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223– 233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by examining deeper such connections. A key ingredient of our study is the introduction of a new equivalent relation between two dust-like self-similar sets called matchable condition. Thanks to a certain measure-preserving property of bi-Lipschitz maps between dust-like self-similar sets, we show that the matchable condition is a necessary condition for Lipschitz equivalence. Using the matchable condition we prove several conditions on the Lipschitz equivalence of dust-like self-similar sets based on the algebraic properties of the contraction ratios, which include a complete characterization of Lipschitz equivalence when the multiplication groups generated by the contraction ratios have full rank. We also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e. they are generated by IFS with two contractive similarities). Some other results are also presented, including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio.

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP-approach to the immersion problem of P^m. A form of the Euler class viewpoint is applied to show TC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P^3., updated, final version to appear in Transactions of the American Mathematical Society

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schutzenberger) could be used to dene a new basis for the ring of quasisymmetric functions we call \Quasisymmetric Schur functions" (QS functions for short). In this paper we develop the combinatorics of these polynomials futher, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coecients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis. A composition (weak composition) with n parts is a sequence of n positive (nonnegative) integers, respectively. A partition is a composition whose parts are monotone nonincreasing. If is a weak composition, composition, or partition, we let '( ) denote the number of parts of . Throughout this article is a weak composition with '( ) = n while and denote compositions and partitions, respectively, with '( ) n, '( ) n. The polynomials in this paper (Schur functions, Demazure atoms and characters, QS functions) depend on a

In an earlier paper we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, including all connected symmetric pairs over C. In this paper we show that the pairs (GL(V),O(V)), (GL(V),U(V)), (U(V), O(V)), (O(V ⊕ W), O(V) x O(W)), (U(V ⊕ W), U(V) × U(W)) are regular, where V and W are quadratic or Hermitian spaces over an arbitrary local field of characteristic zero. We deduce from this that the pairs (GL n (ℂ), O n (ℂ)) and (O n+m (ℂ),O n (ℂ) × O m (ℂ)) are Gelfand pairs.

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V \choose r}$ ($:= \{\mbox{$r$-subsets of $V$}\}$). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ is ${\mathcal H}$ likely to contain a perfect matching (that is, $n/r$ disjoint $r$-sets)? In 2008 Johansson, Vu and the author showed that this is true for $M>C_rn\log n$. More recently the author proved the asymptotically correct version of that result: for fixed $C> 1/r$ and $M> Cn\log n$, $P({\mathcal H} ~\mbox{contains a perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}.$ The present work completes a proof, begun in that recent paper, of the definitive "hitting time" statement: $\mbox{Theorem.}$ If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal K}$, ${\mathcal H}_t=\{A_1\dots A_t\}$, and \[ T=\min\{t:A_1\cup \cdots\cup A_t=V\}, \] then $P({\mathcal H}_T ~\mbox{contains a perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}$.

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled., Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinberg

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that v satisfies δn≤ 1. We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius αn.

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.

This paper adresses the following problem: Given a closed orientable three-manifold M, are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincare-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold M1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of M.

The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlos, Major, and Tusnady. To make the paper self-contained, we include a proof of the approximation result that we need.

Let ( R , m ) (R,\mathfrak {m}) be an n n -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an m \mathfrak {m} -primary ideal a \mathfrak {a} of R R , the relationship between the singularities of the scheme defined by a \mathfrak {a} and those defined by the multiplier ideals J ( a c ) \mathcal {J}(\mathfrak {a}^c) , with c c varying in Q + \mathbb {Q}_+ , are quantified in this paper by showing that the Samuel multiplicity of a \mathfrak {a} satisfies e ( a ) ≥ ( n + k ) n / c n e(\mathfrak {a}) \ge (n+k)^n/c^n whenever J ( a c ) ⊆ m k + 1 \mathcal {J}(\mathfrak {a}^c) \subseteq \mathfrak {m}^{k+1} . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustaţǎ and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$. In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$.

Let G G be a finite simple connected graph on [ n ] [n] and \[ R = K [ x 1 , … , x n ] R = K[x_1, \ldots , x_n] \] the polynomial ring in n n variables over a field K K . The edge ideal of G G is the ideal I ( G ) I(G) of R R which is generated by those monomials x i x j x_ix_j for which { i , j } \{i, j\} is an edge of G G . In the present paper, the possible tuples \[ ( n , d e p t h ( R / I ( G ) ) , r e g ( R / I ( G ) ) , dim ⁡ R / I ( G ) , deg ⁡ h ( R / I ( G ) ) ) , (n, depth(R/I(G)), reg(R/I(G)), \dim R/I(G), \deg h(R/I(G))), \] where deg ⁡ h ( R / I ( G ) ) \deg h(R/I(G)) is the degree of the h h -polynomial of R / I ( G ) R/I(G) , arising from Cameron–Walker graphs on [ n ] [n] will be completely determined.

Given a real number x > 0 x>0 , we determine q s ( x ) ≔ inf ⁡ U ( x ) q_s(x)≔\operatorname {inf}{\mathscr {U}}(x) , where U ( x ) {\mathscr {U}}(x) is the set of all bases q ∈ ( 1 , 2 ] q\in (1,2] for which x x has a unique expansion of 0 0 ’s and 1 1 ’s. We give an explicit description of q s ( x ) q_s(x) for several regions of x x -values. For others, we present an efficient algorithm to determine q s ( x ) q_s(x) and the lexicographically smallest unique expansion of x x . We show that the infimum is attained for almost all x x , but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function q s q_s is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of q s q_s . A large part of the paper is devoted to the level sets L ( q ) ≔ { x > 0 : q s ( x ) = q } L(q)≔\{x>0:q_s(x)=q\} . We show that L ( q ) L(q) is finite for almost every q q , but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant q K L = min ⁡ U ( 1 ) ≈ 1.787 q_{KL}=\operatorname {min}{\mathscr {U}}(1)\approx 1.787 we prove that L ( q K L ) L(q_{KL}) has both infinitely many left- and infinitely many right accumulation points.

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal $I$ of $R$ has linear growth of primary decompositions, then tight closure (of $I$) commutes with localization at the powers of a single element. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius powers of $I$ would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of $I$) commutes with localization at an arbitrary multiplicatively closed subset of $R$. Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of $I$ has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of $R$, strategies for showing that tight closure (of a specified ideal $I$ of $R$) commutes with localization at an arbitrary multiplicatively closed subset of $R$ and for showing that the union of the associated primes of the tight closures of the Frobenius powers of $I$ is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new., Comment: This is to appear in the Transactions of the American Mathematical Society

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.

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.

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.

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

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

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.

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.

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 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

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.

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 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

Motivated by the Blaschke-Santalo inequality, we define for a convex body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52