Journal•ISSN: 1558-8599
Pure and Applied Mathematics Quarterly
International Press of Boston, Inc.
About: Pure and Applied Mathematics Quarterly is an academic journal published by International Press of Boston, Inc.. The journal publishes majorly in the area(s): Moduli space & Conjecture. It has an ISSN identifier of 1558-8599. Over the lifetime, 614 publications have been published receiving 8570 citations. The journal is also known as: PAMQ.
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TL;DR: In this article, the authors show how the beautiful ideas of Tits were developed in our joint works with G. Margulis [MS1], [MS2] and [MS3].
Abstract: Note that from the definition immediately follows that the group G is a free group with free generators gi, i ∈ I. Indeed, let g = g1 i1 . . . g mk ik be any reduced word. Take p ∈ D0, then gp ∈ D± i1 ⊆ H2 \D0. Therefore g 6= 1. One of the main purposes of the present work is to show how the beautiful ideas of Tits were developed in our joint works with G. Margulis [MS1], [MS2], [MS3]. Our interest to free subgroups of linear groups was initiated by the following Problem 1 (V. Platonov) Does there exist a maximal subgroup of infinite index
167 citations
TL;DR: The local-global compatibility conjecture as mentioned in this paper is a conjecture on the multiplicities with which certain GQ-representations appear in the E-vector space of a topological manifold.
Abstract: 1.1. The local-global compatibility conjecture. Fix a prime p, as well as a finite extension E of Qp. If K is an open subgroup of GL2(Ẑ) (referred to as a “tame level”), then one can define a certain E-Banach space Ĥ(K)E , equipped with an action of GQ × GL2(Qp), by taking the inductive limit of the étale cohomology with coefficients in E of the modular curves of arbitrary p-power level and of tame level K, and then completing with respect to the norm induced by the OE-submodule of integral cohomology classes. Passing to the locally convex inductive limit over all tame levels K, we obtain a complete locally convex topological E-vector space ĤE equipped with a representation of GQ × GL2(Af ) that is (so to speak) “smooth in the prime-to-p-directions, but unitary Banach in the p-adic direction”. (See Subsection 7.2 for the precise definitions of these various topological vector spaces.) The object of this note is to explain a conjecture on the multiplicities with which certain GQ-representations appear in ĤE . This conjecture is in some sense the most optimistic possible, in light of what is already known, or believed, to be true. In order to state the conjecture, we must first admit the truth of a “local padic Langlands conjecture for GL2”. The idea that such a conjecture should (or even could) exist is due largely to Breuil, and has been extensively developed both by him and others. In what follows, we will take as given the most optimistic version of this conjecture, namely that to any continuous representation of GQp on a two dimensional E-vector space V there is associated in a natural manner an
137 citations
TL;DR: In this article, Buchstaber, Panov, and Baskakov studied the cohomology ring, homotopy groups, and triple Massey products of a moment-angle simplicial complex.
Abstract: Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.
136 citations
TL;DR: In this paper, a survey of maximal representations of homomorphic Lie groups is presented, which is the subset of Hom(Γg,G ) characterized by the maximality of the Toledo invariant.
Abstract: Let G be a connected semisimple Lie group such that the associ- ated symmetric space X is Hermitian and let Γg be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the study of maximal representations of Γg into G, that is the subset of Hom(Γg ,G ) characterized by the maximality of the Toledo invariant ((17) and (15)). Then we concen- trate on the particular case G =S p(2n, R), and we show that if ρ is any maximal representation then the image ρ(Γg) is a discrete, faithful realiza- tions of Γg as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.
125 citations
TL;DR: A weakly holomorphic modular form of weight k ∈ 2Z for Γ = PSL2(Z) is a holomorphic function f on the upper half-plane that satisfies f( cτ+d ) = (cτ + d)f(τ) for all (a b c d ) ∈ Γ and that has a q-expansion of the form f(τ), where q = e and n 0 = ord∞(f).
Abstract: For this paper we assume familiarity with the basics of the theory of modular forms as may be found, for instance, in Serre’s classic introduction [12]. A weakly holomorphic modular form of weight k ∈ 2Z for Γ = PSL2(Z) is a holomorphic function f on the upper half-plane that satisfies f( cτ+d ) = (cτ + d)f(τ) for all ( a b c d ) ∈ Γ and that has a q-expansion of the form f(τ) = ∑ n≥n0 a(n)q , where q = e and n0 = ord∞(f). Such an f is holomorphic if n0 ≥ 0 and a cusp form if n0 ≥ 1. Let Mk denote the vector space of all weakly holomorphic modular forms of weight k. Any nonzero f ∈ Mk satisfies the valence formula
121 citations