Academic literature on the topic 'Geometric finiteness'

Cette thèse est consacrée à l’étude des orbivariétés projectives convexes géométriquement finies, et fait suite aux travaux de Ballas, Cooper, Crampon, Leitner, Long, Marquis et Tillmann sur le sujet. Une orbivariété projective convexe est le quotient d’un ouvert convexe et borné d’une carte affine de l’espace projectif réel (appelé aussi ouvert proprement convexe) par un groupe discret de transformations projectives préservant cet ouvert. S’il n’y a pas de segment dans le bord du convexe, on dit que l’orbivariété est strictement convexe, et si de plus il y a un unique hyperplan de support en chaque point du bord, on dit qu’elle est ronde. Suivant Cooper-Long-Tillmann et Crampon-Marquis, on dit qu’une orbivariété strictement convexe est géométriquement finie si son cœur convexe est l’union d’un compact et d’un nombre fini de bouts, appelés pointes, où le rayon d’injectivité est inférieur à une constante ne dépendant que de la dimension. Comprendre la géométrie des pointes est primordial pour l’étude des orbivariétés géométriquement finies. Dans le cas strictement convexe, la seule restriction connue sur l’holonomie des pointes vient d’une généralisation du lemme de Margulis due à Cooper-Long-Tillmann et Crampon-Marquis, qui implique que cette holonomie est virtuellement nilpotente. On donne dans cette thèse une caractérisation de l’holonomie des pointes des orbivariétés strictement convexes et des orbivariétés rondes. En généralisant la méthode de Cooper, qui a produit le seul exemple connu jusqu’ici d’une pointe de variété strictement convexe dont l’holonomie n’est pas virtuellement abélienne, on construit des pointes de variétés strictement convexes et de variétés rondes dont l’holonomie est isomorphe à n’importe quel groupe nilpotent sans torsion de type fini. En collaboration avec M. Islam et F. Zhu, on démontre que dans le cas des groupes relativement hyperboliques sans torsion, les représentations relativement P1-anosoviennes (au sens de Kapovich-Leeb, Zhu et Zhu-Zimmer) qui préservent un ouvert proprement convexe sont exactement les holonomies des variétés rondes géométriquement finies.Dans le cas des orbivariétés projectives convexes non strictement convexes, il n’y a pas pour l’instant de définition satisfaisante de la finitude géométrique. Toutefois, Cooper-Long-Tillmann puis Ballas-Cooper-Leitner ont proposé une définition de pointe généralisée dans ce contexte. Bien qu’ils demandent que l’holonomie des pointes généralisées soit virtuellement nilpotente, tous les exemples connus jusqu’à présent avaient une holonomie virtuellement abélienne. On construit des exemples de pointes généralisées dont l’holonomie peut être n’importe quel groupe nilpotent sans torsion de type fini. On s’autorise également à modifier la définition originale de Cooper-Long-Tillmann en affaiblissant l’hypothèse de nilpotence en une hypothèse naturelle de résolubilité, ce qui nous permet de construire de nouveaux exemples dont l’holonomie n’est pas virtuellement nilpotente.Une orbivariété géométriquement finie qui n’a pas de pointes, c’est-à-dire dont le cœur convexe est compact, est dite convexe cocompacte. On dispose par les travaux de Danciger-Guéritaud-Kassel d’une définition de la convexe cocompacité pour les orbivariétés projectives convexes sans hypothèse de stricte convexité, contrairement au cas géométriquement fini. Ils démontrent que l’holonomie d’une orbivariété projective convexe convexe cocompacte est Gromov hyperbolique si et seulement si la représentation associée est P1-anosovienne. À l’aide de ce résultat, de la théorie de Vinberg et des travaux d’Agol et Haglund-Wise sur les groupes hyperboliques cubulés, on construit en collaboration avec S. Douba, T. Weisman et F. Zhu des représentations P1-anosoviennes pour tout groupe hyperbolique cubulé. Ceci fournit de nouveaux exemples de groupes hyperboliques admettant des représentations anosoviennes<br>This thesis is devoted to the study of geometrically finite convex projective orbifolds, following work of Ballas, Cooper, Crampon, Leitner, Long, Marquis and Tillmann. A convex projective orbifold is the quotient of a bounded, convex and open subset of an affine chart of real projective space (called a properly convex domain) by a discrete group of projective transformations that preserve it. We say that a convex projective orbifold is strictly convex if there are no non-trivial segments in the boundary of the convex subset, and round if in addition there is a unique supporting hyperplane at each boundary point. Following work of Cooper-Long-Tillmann and Crampon-Marquis, we say that a strictly convex orbifold is geometrically finite if its convex core decomposes as the union of a compact subset and of finitely many ends, called cusps, all of whose points have an injectivity radius smaller than a constant depending only on the dimension. Understanding what types of cusps may occur is crucial for the study of geometrically finite orbifolds. In the strictly convex case, the only known restriction on cusp holonomies, imposed by a generalization of the celebrated Margulis lemma proven by Cooper-Long-Tillmann and Crampon-Marquis, is that the holonomy of a cusp has to be virtually nilpotent. We give a complete characterization of the holonomies of cusps of strictly convex orbifolds and of those of round orbifolds. By generalizing a method of Cooper, which gave the only previously known example of a cusp of a strictly convex manifold with non virtually abelian holonomy, we build examples of cusps of strictly convex manifolds and round manifolds whose holonomy can be any finitely generated torsion-free nilpotent group. In joint work with M. Islam and F. Zhu, we also prove that for torsion-free relatively hyperbolic groups, relative P1-Anosov representations (in the sense of Kapovich-Leeb, Zhu and Zhu-Zimmer) that preserve a properly convex domain are exactly the holonomies of geometrically finite round manifolds.In the general case of non strictly convex projective orbifolds, no satisfactory definition of geometric finiteness is known at the moment. However, Cooper-Long-Tillmann, followed by Ballas-Cooper-Leitner, introduced a notion of generalized cusps in this context. Although they only require that the holonomy be virtually nilpotent, all previously known examples had virtually abelian holonomy. We build examples of generalized cusps whose holonomy can be any finitely generated torsion-free nilpotent group. We also allow ourselves to weaken Cooper-Long-Tillmann’s original definition by assuming only that the holonomy be virtually solvable, and this enables us to construct new examples whose holonomy is not virtually nilpotent.When a geometrically finite orbifold has no cusps, i.e. when its convex core is compact, we say that the orbifold is convex cocompact. Danciger-Guéritaud-Kassel provided a good definition of convex cocompactness for convex projective orbifolds that are not necessarily strictly convex. They proved that the holonomy of a convex cocompact convex projective orbifold is Gromov hyperbolic if and only if the associated representation is P1-Anosov. Using these results, Vinberg’s theory and work of Agol and Haglund-Wise about cubulated hyperbolic groups, we construct, in collaboration with S. Douba, T. Weisman and F. Zhu, examples of P1-Anosov representations for any cubulated hyperbolic group. This gives new examples of hyperbolic groups admitting Anosov representations

A group Γ is of type F_n for some n ≥ 1 if it has a classifying complex with finite n-skeleton. These properties generalise the classical notions of finite generation and finite presentability. We investigate the higher finiteness properties for fibre products of groups.

In this paper, we describe various definitions of geometrical finiteness for discrete hyperbolic groups in any dimension, and prove their equivalence. This generalises what has been worked out in two and three dimensions by Marden. Beardon, Maskit, Thurston and others. We also discuss the nature of convex fundamental domains for such groups. We begin the paper with a discussion of results related to the Margulls Lemma and Bieberbach Theorems.

Of many modern constructions in geometry Calabi Yau manifolds hold special relevance in theoretical physics. These manifolds naturally arise from the study of compactification of certain string theories. In particular Calabi Yau manifolds of dimension three, commonly known as threefolds, are widely used for compactifications of heterotic string theories. Among the many constructions, that of complete intersection Calabi Yau manifolds (CICY) is generally regarded to be the simplest. Furthermore, CICY threefolds have been proven to exist only in finite number. In the following text CICY manifolds will be analyzed, with particular attention to threefolds. A general description of some of their topological quantities and their calculation is offered. Lastly, a proof of the finiteness of CICY threefolds is given.

Cette thèse est consacrée à l'étude des variétés projectives strictement convexes de volume fini. Une telle variété est le quotient G\U d'un ouvert proprement convexe U de l'espace projectif réel RP^(n-1) par un sous-groupe discret sans torsion G de SLn(R) qui préserve U. Dans un premier temps, on étudie l'adhérence de Zariski des holonomies de variétés projectives strictement convexes de volume fini. Pour une telle variété G\U, on montre que, soit G est Zariski-dense dans SLn(R), soit l'adhérence de Zariski de G est conjuguée à SO(1,n-1). On s'intéresse ensuite à l'espace des modules des structures projectives strictement convexes de volume fini. On montre en particulier que cet espace des modules est un fermé de l'espace des représentations<br>In this thesis, we study strictly convex projective manifolds of finite volume. Such a manifold is the quotient G\U of a properly convex open subset U of the real projective space RP^(n-1) by a discrete torsionfree subgroup G of SLn(R) preserving U. We study the Zariski closure of holonomies of convex projective manifolds of finite volume. For such manifolds G\U, we show that either the Zariski closure of G is SLn(R) or it is a conjugate of SO(1,n-1).We also focuss on the moduli space of strictly convex projective structures of finite volume. We show that this moduli space is a closed set of the representation space

碩士<br>國立臺灣師範大學<br>數學系<br>98<br>In these paper, we consider several properties of Normal Projection Energy. Firstly, among the class of $C^{1,1}$-smooth knots, the upper bound of Normal Projection Energy gives a uniform lower bound of Gromov's distorsion of knots. Secondly, Normal Projection Energy is bounded by the product of total curvature and ropelength. Thirdly, to prove the bound of Normal Projection Energy, we study the curves which attain the infimum of the total absolute curvature in the set of curves contained in a ball with fixed endpoints and length.

Wu, Lijiang.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.<br>Includes bibliographical references (leaves 102-105).<br>Abstracts in English and Chinese.<br>Chapter 0 --- Introduction --- p.6<br>Chapter 1 --- Background knowledge --- p.9<br>Chapter 1.1 --- Comparison theorems --- p.9<br>Chapter 1.2 --- Bochner techniques --- p.13<br>Chapter 1.3 --- Eigenvalue estimates for Laplacian operator --- p.14<br>Chapter 1.4 --- Spectral theory for Schrodinger operator on Rieman- nian manifolds --- p.16<br>Chapter 2 --- Vanishing theorems --- p.20<br>Chapter 2.1 --- Liouville type theorem for Lp subharmonic functions --- p.20<br>Chapter 2.2 --- Generalized type of vanishing theorem --- p.21<br>Chapter 3 --- Finite dimensionality results --- p.28<br>Chapter 3.1 --- Three types of integral inequalities --- p.28<br>Chapter 3.2 --- Weak Harnack inequality --- p.34<br>Chapter 3.3 --- Li's abstract finite dimensionality theorem --- p.37<br>Chapter 3.4 --- Applications of the finite dimensionality theorem --- p.42<br>Chapter 4 --- Ends of Riemannian manifolds --- p.48<br>Chapter 4.1 --- Green's function --- p.48<br>Chapter 4.2 --- Ends and harmonic functions --- p.53<br>Chapter 4.3 --- Some topological applications --- p.72<br>Chapter 5 --- Splitting theorems --- p.79<br>Chapter 5.1 --- Splitting theorems for manifolds with non-negative Ricci curvature --- p.79<br>Chapter 5.2 --- Splitting theorems for manifolds of Ricci curvature with a negative lower bound --- p.83<br>Chapter 5.3 --- Manifolds with the maximal possible eigenvalue --- p.93<br>Bibliography --- p.102

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

This paper brings to the attention of the AES community our experience with developing and trying out a new course that focuses on the generation of system configuration (geometry, architecture, and drawing) during the optimization of performance. The configuration is free to morph. Real systems are destined to remain imperfect because of finiteness constraints. They are plagued by resistances to the flow of fluid, heat, and electricity. Resistances are always finite because of constraints. The balancing and distributing of resistances (irreversibility) through the available volume is the mechanism that generates the architecture. Our approach to this key topic is evolutionary, and from simple to complex. In the 2004 format, the course covers several main topics: the relationship between optimization and the generation of configuration, multi-scale hierarchical structures for fluid flow and heat flow, and tree-shaped networks for collection and distribution. The paper relies on classroom-level examples to illustrate these topics. The paper also discusses the teaching method, and the applicability of this course concept to other domains (cost minimization, urban design, etc.).

This work, the first of two parts, presents the development of a new analytic solution of acoustic scattering and/or radiation by arbitrary bodies of revolution under heavy fluid loading. The approach followed is the construction of a three-dimensional Wiener-Hopf technique with Fourier transforms that operate on the finite object’s arclength variable (the object’s practical finiteness comes about, in a Wiener-Hopf sense, by formally bringing to zero the radius of its semi-infinite generator curve for points beyond a prescribed station). Unlike in the classical case of a planar semi-infinite geometry, the kernel of the integral equation is non-translational and therefore with independent wavenumber spectra for its receiver and source arclengths. The solution procedure begins by applying a symmetrizing spatial operator that reconciles the regions of (+) and (−) analyticity of the kernel’s two-wavenumber transform with those of the virtual sources. The spatially symmetrized integral equation is of the Fredholm 2nd kind and thus with a strong unit “diagonal” — a feature that makes possible the Wiener-Hopf factorization of its transcendental doubly-transformed kernel via secondary spectral manipulations. The companion paper [1] will present a numerical demonstration of the new analysis for canonical problems of fluid-structure interaction for finite bodies of revolution.