Search

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.

Inspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(\mu)$ and prove that its dual space is the known space ${\rm RBMO}(\mu)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(\mu)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$., Comment: This paper has been withdrawn by the authors, since it has already been published

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$ T. W. Cusick conjectured that ct > 1/2. We have the elementary bound 0 < ct < 1; however, no bound of the form 0 < α ≤ ct or ct ≤ β < 1, valid for all t, is known. In this paper, we prove that ct > 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang-Mills gauge theory from twisting by the Picard group of a four manifold in the case when the fundamental group is free abelian. We then generalize it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic four manifolds., Comment: typos fixed, Rewrite Section 4 to reduce technicality

This paper deals with the category of paracompact locales (‘pointless topologies’), defined in the classic paper [6] of Isbell. A full discussion concerning paracompact locales can be found in Dowker and Strauss[2] and in Pultr[10, 11]. We shall provide a description of paracompact Tychonoff locales by means of a system of suitably chosen metric spaces. This answers Pultr's question whether each paracompact Tychonoff locale is a closed sublocale of a (localic) product of metric spaces.

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case, orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather, they may equidistribute on a fractal subset. Understanding the behaviour of these orbits near the boundary is central to Patterson-Sullivan theory and much further work. Our theorem characterizes the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish an formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which was not known previously even in the lattice case., 33 pages, 1 figure; Accepted for Publication in Mathematical Proceedings of the Cambridge Philosophical Society

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin–Löf Type Theory, in the presence of Voevodsky’s Univalence Axiom.In his 1978 paper, “the type theoretic interpretation of constructive set theory” Aczel introduced a model of constructive set theory in type theory, using a W-type quantifying over a universe, and an inductively defined equivalence relation on it. Our investigation takes this W-type and instead considers the identity type on it, which can be computed from the univalence axiom. Our thesis is that this gives a model of multisets. In order to demonstrate this, we adapt axioms of constructive set theory to multisets, and show that they hold for our model.

RésuméWe study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ y ⩽ x and on average over q ⩽ x/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations., 24 pages. Comments are welcome! The ancillary file is a longer version with some more details and alternative proofs (compiled from the same source). The version on http://www.cip.ifi.lmu.de/~grinberg/algebra/ will probably get quicker updates. v3: Minor corrections

In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complexL-series for the family of quadratic twists of all optimal elliptic curves over${\mathbb Q}$.

Denote ΘCas the Frobenius class of a curveCover the finite field 𝔽q. In this paper we determine the expected value of Tr(ΘCn) whereCruns over all biquadratic curves whenqis fixed andgtends to infinity. This extends work done by Rudnick [15] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.

In this paper, we study families of singular surfaces in ℂ3 parametrised by $\mathcal {A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (ℂ2, 0) → (ℂ3, 0). We investigate the following question: topological triviality implies Whitney equisingularity of the unfolding F? We provide a complete answer to this question, by giving counterexamples showing how the conjecture can be false.

Amicable pairs for a fixed elliptic curve defined over $\mathbb{Q}$ were first considered by Silverman and Stange where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones to give a precise asymptotic constant. The author previously proved an upper bound for the average number of amicable pairs over the family of all elliptic curves. In this paper we improve this result to an asymptotic for the average number of amicable pairs for a family of elliptic curves., Comment: 27 pages, with an appendix by Sumit Giri

LetKbe an algebraically closed field and$\mathbb{A}$n≅Knaffinen-space. It is known that a finite group$\frak{G}$can only act freely on$\mathbb{A}$nifKhas characteristicp> 0 and$\frak{G}$is ap-group. In that case the group action is “non-linear” and the ring of regular functionsK[$\mathbb{A}$n] must be atrace-surjectiveK−$\frak{G}$-algebra.Now letkbe an arbitrary field of characteristicp> 0 and letGbe a finitep-group. In this paper we study the category$\mathfrak{Ts}$of all finitely generated trace-surjectivek−Galgebras. It has been shown in [13] that the objects in$\mathfrak{Ts}$are precisely those finitely generatedk−GalgebrasAsuch thatAG≤Ais a Galois-extension in the sense of [7], with faithful action ofGonA. Although$\mathfrak{Ts}$is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show thats-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category$\mathfrak{Ts}$also has “weakly initial objects”, which are closely related to the essential dimension ofGoverk. Our results yield a geometric structure theorem for free actions of finitep-groups on affinek-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory ofp-groups and, potentially, to a new constructive approach to homogeneous invariant theory.

Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a coloured tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author.

LetSbe a polynomial ring over a fieldKof characteristic zero and letM⊂Sbe an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case whereMis a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime idealP⊂ScontainingMbe such that the localisationMPhasnon-maximal analytic spread. Our technique describes, in fact, a concrete obstruction forPto be an asymptotic prime divisor ofMwith respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes.

Let $S(t) = \tfrac{1}{\pi} \arg \zeta (\frac12 + it)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its iterates \begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,{\rm d}\tau\, + \delta_n\,, \end{equation*} where $\delta_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that $S_n(t) = O(\log t/ (\log \log t)^{n+1})$. The order of magnitude of this estimate was never improved up to this date. The best bounds for $S(t)$ and $S_1(t)$ are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate \begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*} for all $n\geq 2$, with the constants $C_n^{\pm}$ decaying exponentially fast as $n \to \infty$. This improves (for all $n \geq 2$) a result of Wakasa, who had previously obtained such bounds with constants tending to a stationary value when $n \to \infty$. Our method uses special extremal functions of exponential type derived from the Gaussian subordination framework of Carneiro, Littmann and Vaaler for the cases when $n$ is odd, and an optimized interpolation argument for the cases when $n$ is even. In the final section we extend these results to a general class of $L$-functions., Comment: 23 pages. To appear in Math. Proc. Cambridge Philos. Soc

For a group G first order definable in a structure M, we continue the study of the “definable topological dynamics” of G (from [9] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below.Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G*/(G*)000M of G, which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G*/(G*)000M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in SG(M) are definable; (ii) the kernel of the surjective homomorphism from G*/(G*)000M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th(M) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable.In the situation when all types in SG(M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space Σ E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarn\'ik are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarn\'ik type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarn\'ik type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves., Comment: 24 pages. Updated to include bibliography, which was incorrectly compiled initially

A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

Let E/Q be an elliptic curve, p a prime and K∞/K the anticyclotomic Zp-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has supersingular reduction at p.

The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\mathbb Z^n,G)$, with the order of the Weyl group inverted., Comment: 27 pages, with an appendix by Vic Reiner

The category of rational SO(2)-equivariant spectra admits an algebraic model. That is, there is an abelian category A(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)-equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra? The category A(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non-Noetherian ring. Hence the standard techniques for constructing a monoidal model structure cannot be applied. In this paper we construct a monoidal model structure on A(SO(2)) and show that the derived product on the homotopy category is compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K_p-local stable homotopy category. We also provide a monoidal Quillen equivalence to a simpler monoidal model category R_*-modules that has explicit generating sets. Having monoidal model structures on A(SO(2)) and R_*-modules removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)-equivariant spectra., 22 Pages. arXiv admin note: text overlap with arXiv:1201.6610

In this paper, we prove a Tb theorem on product spaces $\mathbb{R}$n × $\mathbb{R}$m, where b(x1, x2) = b1(x1)b2(x2), b1 and b2 are para-accretive functions on $\mathbb{R}$n and $\mathbb{R}$m, respectively.

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic elements., Comment: 9 pages, 7 figures

Let HX be the trigraded Hilbert function of a set X of reduced points in $\mathbb{P}$1 × $\mathbb{P}$1 × $\mathbb{P}$1. We show how to extract some geometric information about X from HX. This paper generalises a similar result of Giuffrida, Maggioni and Ragusa about sets of points in $\mathbb{P}$1 × $\mathbb{P}$1.

The coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed by Steinberg, Lusztig and Beynon-Lusztig. In this paper we formulate a notion of spin coinvariant algebra for every Weyl group. Then we compute all the spin fake degrees for each classical Weyl group, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in the spin coinvariant algebra. The spin fake degrees for the exceptional Weyl groups are given in a sequel., Comment: v2, 39 pages, title modified (with "of classical type" added), the original version was split into two parts following editor's suggestion; this v2 is the part one (to appear in Math. Proc. Cambridge Philos. Soc.), with a sequel dealing with the exceptional type

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two., Comment: 35 pages, 2 figures

Assume $f{:}\,U\subset \C\to \C$ is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| ≤ 1. If λ is not a root of unity, there is a formal power series φf(z) = z + ${\cal O}$(z2) such that φf(λ z) = f(φf(z)). This power series is unique and we denote by Rconv(f) ∈ [0,+∞] its radius of convergence. We denote by Rgeom(f) the largest radius r ∈ [0, Rconv(f)] such that φf(D(0,r)) ⊂ U. In this paper, we present new elementary techniques for studying the maps f ↦ Rconv(f) and f ↦ Rgeom(f). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work., 17 pages, to appear in Math. Proc. Camb. Philos. Soc

In this paper we have two objectives. The first is to investigate the restriction of a metaplectic cover to an arbitrary torus in GL2. This will be explained at greater length below, and the main results are Theorems 1 and 2. The second is an application of the same ideas to introduce the arithmetic function P, which has already appeared in a special case in [9], and to prove the fundamental property given by Theorem 3. These theorems will be proved in §§ 2 and 3. In §§ 4 and 5 we remark on the appearence of the function P in the formula of Loxton and Matthews [5], [6] for the biquadratic Gauss sum and discuss the structure of this formula.

This paper studies coefficientsyh, nof sequences of polynomialsdefined by non-linear recurrences. A typical example to which the results of this paperapply is that of the sequencewhich arises in the study of binary trees. For a wide class of similar sequences a general distribution law for the coefficientsyh, nas functions ofnwithhfixed is established. It follows from this law that in many interesting cases the distribution is asymptotically Gaussian near the peak. The proof relies on the saddle point method applied in a region where the polynomials grow doubly exponentially ash→ ∞. Applications of these results include enumerations of binary trees and 2–3 trees. Other structures of interest in computer science and combinatorics can also be studied by this method or its extensions.