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Analysts’ functions are divided into discovery and interpretation roles, but distinguishing between the two is nontrivial. We conjecture that analysts’ interpretation skill can be gauged by their forecast revisions following material unanticipated news, in particular, following nonearnings 8-K reports, which arrive at the market unexpectedly. We establish that unanticipated 8-Ks are informative for analysts and find that analysts who are more likely to revise their forecasts following unanticipated 8-Ks provide more timely and accurate forecasts. We document a positive association between analysts’ tendency to react to unanticipated 8-Ks and market reaction to their recommendation changes, suggesting investors prefer these analysts’ opinions.
In this book, three authors introduce readers to strong approximation methods, analytic pro-p groups and zeta functions of groups. Each chapter illustrates connections between infinite group theory, number theory and Lie theory. The first introduces the theory of compact p-adic Lie groups. The second explains how methods from linear algebraic groups can be utilised to study the finite images of linear groups. The final chapter provides an overview of zeta functions associated to groups and rings. Derived from an LMS/EPSRC Short Course for graduate students, this book provides a concise introduction to a very active research area and assumes less prior knowledge than existing monographs or original research articles. Accessible to beginning graduate students in group theory, it will also appeal to researchers interested in infinite group theory and its interface with Lie theory and number theory.
In September 2007, the London Mathematical Society and the EPSRC sponsored a ‘short course for graduates’ in Oxford, under the heading ‘Asymptotic methods in infinite group theory’. This was organised by Dan Segal and consisted of three series of lectures. The present book is basically a record of these lectures, somewhat polished and expanded. It is intended to serve as an introduction, for beginning research students and for interested non-specialists, to some areas of current activity in algebra: the questions mostly originate in group theory but the methodology encompasses a wide range of mathematics, involving topology, algebraic geometry, number theory and combinatorics.
The theory of Lie groups is highly developed and of relevance in many parts of contemporary mathematics and theoretical physics. Loosely speaking, a Lie group is a group with the additional structure of a real differentiable manifold, given by local coordinate systems, such that the group operations are smooth functions.
Historically, the study of Lie groups, over the real and complex numbers, arose toward the end of the 19th century, from the analysis of continuous symmetries of differential equations by the mathematician Sophus Lie and others. Around the middle of the 20th century, mathematicians such as Armand Borel and Claude Chevalley found that many of the foundational results concerning Lie groups could be developed completely algebraically, giving rise to the theory of algebraic groups defined over arbitrary fields. This insight opened the way for entirely new directions of investigation. Much of the theory of p-adic Lie groups was developed in the 1960s by mathematicians such as Nicolas Bourbaki, Michel Lazard and Jean-Pierre Serre. Since then the study of p-adic Lie groups and analogues of Lie groups over adele rings has largely been motivated by questions from number theory, e.g. regarding automorphic forms and Galois representations. More recently, p-adic Lie groups have also become a key tool in infinite group theory.
Throughout, let p be a prime. The real numbers ℝ form a completion of the rational numbers ℚ. Similarly, the field of p-adic numbers ℚp is obtained by completing ℚ, albeit with respect to a different, non-archimedean notion of distance.
From a purely algebraic point of view, there is not a lot one can say about infinite groups in general. Traditionally, these have been studied to good effect in combination with topology or geometry. These lectures represent an introduction to some recent developments that arise out of looking at infinite groups from a point of view inspired – in a general sense – by number theory; specifically the interaction between ‘local’ and ‘global’, where by ‘local’ properties of a group G, in this context, one means the properties of its finite quotients, or equivalently properties of its profinite completion Ĝ. The second chapter directly addresses the interplay between certain finitely generated groups and their finite images. The other two chapters are more specifically ‘local’ in emphasis: Chapter I concerns the algebraic structure of certain pro-p groups, while Chapter III introduces a way of studying the rich arithmetical data encoded in certain infinite groups and related structures.
A motivating example for all of the above is the question of ‘subgroup growth’. Say G has sn(G) subgroups of index at most n for each n; the function n ↦ sn(G) is the subgroup growth function of G, and is finite-valued if we assume that G is finitely generated. Now we can ask (inspired perhaps by Gromov's celebrated polynomial growth theorem): what does it mean for the global structure of a finitely generated group if its subgroup growth function is (bounded by a) polynomial?
Here, algebraic group will mean a Zariski-closed subgroup of SLn (K), for some n ∈ ℕ and some algebraically closed field K. For background and terminology, see [B2], [H2], [PR] and [W1]. In this section, topological language refers to the Zariski topology.
The following theorem, due to Merzljakov, in a sense provides the philosophical background to all the ellipticity results concerning groups of Lie type; a sharper result specific to simple groups is stated below.
Theorem 5.1.1 [M2] Every algebraic group is verbally elliptic.
This depends on Chevalley's concept of constructible sets. Let V = Kd be the affine space, with its Zariski topology. A subset of V is constructible if it is a finite union of sets of the form C ∩ U where C is closed and U is open. A morphism from V to V1 = Kl is a mapping defined by l polynomials.
The key result is
Proposition 5.1.2 Let Y be a constructible subset of V, with closure Ȳ.
(i) (See [W1], 14.9) The set Y contains a subset U that is open and dense in Ȳ.
(ii) (Chevalley, see [H2], §4.4) If f : V → V1 is a morphism then f(Y) is a constructible subset of V1.
After a forty-year lull, the study of word-values in groups has sprung back into life with some spectacular new results in finite group theory. These are largely motivated by applications to profinite groups, including the solution of an old problem of Serre. This book presents a comprehensive account of the known results, both old and new. The more elementary methods are developed from scratch, leading to self-contained proofs and improvements of some classic results about infinite soluble groups. This is followed by a detailed introduction to more advanced topics in finite group theory, and a full account of the applications to profinite groups. The author presents proofs of some very recent results and discusses open questions for further research. This self-contained account is accessible to research students, but will interest all research workers in group theory.