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1
Markham, Sarah. "Hypercomplex hyperbolic geometry." Thesis, Durham University, 2003. http://etheses.dur.ac.uk/3698/.
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The rank one symmetric spaces of non-compact type are the real, complex, quaternionic and octonionic hyperbolic spaces. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still in its infancy. The purpose of this thesis is to investigate the conditions for discrete group action in quaternionic and octonionic hyperbolic 2-spaces and their geometric consequences, in the octonionic case, in terms of lower bounds on the volumes of non-compact manifolds. We will also explore the eigenvalue problem for the 3 x 3 octonionic matrices germane to the Jordan algebra model of the octonionic hyperbolic plane. In Chapters One and Two we concentrate on discreteness conditions in quaternionic hyperbolic 2-space. In Chapter One we develop a quaternionic Jørgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. In Chapter Two we give a generalisation of Shimizu's Lemma to groups of isometries of quaternionic hyperbolic 2-space containing a screw-parabolic element. In Chapter Three we present the Jordan algebra model of the octonionic hyperbolic plane and develop a generalisation of Shimizu's Lemma to groups of isometries of octonionic hyperbolic 2-space containing a parabolic map. We use this result to determine estimates of lower bounds on the volumes of non-compact closed octonionic 2-manifolds. In Chapter Four we construct an octonionic Jørgensen's inequality for non-elementary groups of isometries of octonionic hyperbolic 2-space generated by two elements, one of which is loxodromic. In Chapter Five we solve the real eigenvalue problem Xv = λv, for the 3 x 3 ɸ-Hermitian matrices, X, of the Jordan algebra model of the octonionic hyperbolic plane. Finally, in Chapter Six we consider the embedding of collars about real geodesies in complex hyperbohc 2-space, quaternionic hyperbolic 2-space and octonionic hyperbolic 2-space.
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Marshall, T. H. (Timothy Hamilton). "Hyperbolic Geometry and Reflection Groups." Thesis, University of Auckland, 1994. http://hdl.handle.net/2292/2140.
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The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ≤ k ≤ n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length γ we then derive power series for the volume, both in u = sinγ/2, and in γ itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and pεC). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all π/2, and the remaining dihedral angles are all sub-multiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g≥2, which we conjecture to be of least volume among such manifolds.
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Murray, Marilee Anne. "Hyperbolic Geometry and Coxeter Groups." Bowling Green State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1343040882.
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Walker, Mairi. "Continued fractions and hyperbolic geometry." Thesis, Open University, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.700134.
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This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. This intuitive approach gives insight into the theory of continued fractions that is not so easy to obtain from traditional algebraic methods. Using it, we provide a more extensive study of both Rosen continued fractions and even-integer continued fractions than many previous works, yielding new results, and revisiting classical theorems. We also study two types of complex continued fractions, namely Gaussian integer continued fractions and Bianchi continued fractions. As well as providing a more elegant and simple theory of continued fractions, our approach leads to a natural generalisation of continued fractions that has not been explored, before.
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Thorgeirsson, Sverrir. "Hyperbolic geometry: history, models, and axioms." Thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503.
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Gharamti, Moustafa. "Supersymmetry and geometry of hyperbolic monopoles." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/10479.
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This thesis studies the geometry of hyperbolic monopoles using supersymmetry in four and six dimensions. On the one hand, we show that starting with a four dimensional supersymmetric Yang-Mills theory provides the necessary information to study the geometry of the complex moduli space of hyperbolic monopoles. On the other hand, we require to start with a six dimensional supersymmetric Yang-Mills theory to study the geometry of the real moduli space of hyperbolic monopoles. In chapter two, we construct an off-shell supersymmetric Yang-Mills-Higgs theory with complex fields on three-dimensional hyperbolic space starting from an on-shell supersymmetric Yang-Mills theory on four-dimensional Euclidean space. We, then, show that hyperbolic monopoles coincide precisely with the configurations that preserve one half of the supersymmetry. In chapter three, we explore the geometry of the moduli space of hyperbolic monopoles using the low energy linearization of the field equations. We find that the complexified tangent bundle to the hyperbolic moduli space has a 2-sphere worth of integrable structures that act complex linearly and behave like unit imaginary quaternions. Moreover, we show that these complex structures are parallel with respect to the Obata connection, which implies that the geometry of the complexified moduli space of hyperbolic monopoles is hypercomplex. We also show, as a requirement of analysing the geometry, that there is a one-to-one correspondence between the number of solutions of the linearized Bogomol’nyi equation on hyperbolic space and the number of solutions of the Dirac equation in the presence of hyperbolic monopole. In chapter four and five, we shift the focus to supersymmetric Yang-Mills theories in six dimensional Minkowskian spacetime. Via dimensional reduction we construct a supersymmetric Yang-Mills Higgs theory on R3 with real fields which we then promote to H3. Under certain supersymmetric constraints, we show that hyperbolic monopoles configurations of this theory preserve, again, one half of the supersymmetry. Then, through investigating the geometry of the moduli space we showthat the moduli space is described by real coordinate functions (zero modes), and we construct two sets of 2-sphere of real complex structures that act linearly on the tangent bundle of the moduli space, but don’t behave like unit quaternions. This result coincides with the result of Bielawski and Schwachhöfer, who called this new type of geometry pluricomplex geometry. Finally, we show that in the limiting case, when the radius of curvature H3 is set to infinity, the geometry becomes hyperkähler which is the geometry of the moduli space of Euclidian monopoles.
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Rippy, Scott Randall. "Applications of hyperbolic geometry in physics." CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/1099.
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Naeve, Trent Phillip. "Conics in the hyperbolic plane." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3075.
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An affine transformation such as T(P)=Q is a locus of an affine conic. Any affine conic can be produced from this incidence construction. The affine type of conic (ellipse, parabola, hyperbola) is determined by the invariants of T, the determinant and trace of its linear part. The purpose of this thesis is to obtain a corresponding classification in the hyperbolic plane of conics defined by this construction.
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Bowen, Lewis Phylip. "Density in hyperbolic spaces." Access restricted to users with UT Austin EID Full text (PDF) from UMI/Dissertation Abstracts International, 2002. http://wwwlib.umi.com/cr/utexas/fullcit?p3077409.
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Saratchandran, Hemanth. "Four dimensional hyperbolic link complements via Kirby calculus." Thesis, University of Oxford, 2015. http://ora.ox.ac.uk/objects/uuid:ba72ee75-c22f-4800-a38c-76e5cf411ad9.
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The primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S4 and a smooth standard S2 x S2. We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples.
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Barrett, Benjamin James. "Detecting topological properties of boundaries of hyperbolic groups." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/285572.
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In general, a finitely presented group can have very nasty properties, but many of these properties are avoided if the group is assumed to admit a nice action by isometries on a space with a negative curvature property, such as Gromov hyperbolicity. Such groups are surprisingly common: there is a sense in which a random group admits such an action, as do some groups of classical interest, such as fundamental groups of closed Riemannian manifolds with negative sectional curvature. If a group admits an action on a Gromov hyperbolic space then large scale properties of the space give useful invariants of the group. One particularly natural large scale property used in this way is the Gromov boundary. The Gromov boundary of a hyperbolic group is a compact metric space that is, in a sense, approximated by spheres of large radius in the Cayley graph of the group. The technical results contained in this thesis are effective versions of this statement: we see that the presence of a particular topological feature in the boundary of a hyperbolic group is determined by the geometry of balls in the Cayley graph of radius bounded above by some known upper bound, and is therefore algorithmically detectable. Using these technical results one can prove that certain properties of a group can be computed from its presentation. In particular, we show that there are algorithms that, when given a presentation for a one-ended hyperbolic group, compute Bowditch's canonical decomposition of that group and determine whether or not that group is virtually Fuchsian. The final chapter of this thesis studies the problem of detecting Cech cohomological features in boundaries of hyperbolic groups. Epstein asked whether there is an algorithm that computes the Cech cohomology of the boundary of a given hyperbolic group. We answer Epstein's question in the affirmative for a restricted class of hyperbolic groups: those that are fundamental groups of graphs of free groups with cyclic edge groups. We also prove the computability of the Cech cohomology of a space with some similar properties to the boundary of a hyperbolic group: Otal's decomposition space associated to a line pattern in a free group.
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Heard, Damian. "Computation of hyperbolic structures on 3 dimensional orbifolds /." Connect to theis, 2005. http://eprints.unimelb.edu.au/archive/00001577.
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Raghuvanshi, Anurag. "Particle filter with Hyperbolic Measurements and Geometry Constraints." Ohio University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1366724596.
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Long, Yusen. "Diverse aspects of hyperbolic geometry and group dynamics." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM016.
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Cette thèse explore divers sujets liés à la géométrie hyperbolique et à la dynamique de groupes, dans le but d'étudier l'interaction entre la géométrie et la théorie de groupes. Elle couvre un large éventail de disciplines mathématiques, telles que la géométrie convexe, l'analyse stochastique, la théorie ergodiques et géométriques de groupes, et la topologie en basses dimensions, et cætera. Comme résultats de recherche, la géométrie hyperbolique des corps convexes en dimension infinie est examinée en profondeur, et des tentatives sont faites pour développer la géométrie intégrale en dimension infinie d'un point de vue de l'analyse stochastique. L'étude des gros groupes de difféotopies, un sujet d'actualité en topologie en basses dimensions et en théorie géométrique de groupes, est entreprise avec une détermination complète de leur propriété de point fixe sur les compacts. La thèse étudie la connexité du bord de Gromov des graphes de courbes fins, un outil combinatoire utilisé dans l'étude des groupes d'homéomorphismes des surfaces de type fini. Enfin, la thèse clarifie également certains théorèmes folkloriques concernant les espaces hyperboliques au sens de Gromov et la dynamique des groupes moyennables sur ces espacesThis thesis explores diverse topics related to hyperbolic geometry and group dynamics, aiming to investigate the interplay between geometry and group theory. It covers a wide range of mathematical disciplines, such as convex geometry, stochastic analysis, ergodic and geometric group theory, and low-dimensional topology, etc. As research outcomes, the hyperbolic geometry of infinite-dimensional convex bodies is thoroughly examined, and attempts are made to develop integral geometry in infinite dimensions from a perspective of stochastic analysis. The study of big mapping class groups, a current focus in low-dimensional topology and geometric group theory, is undertaken with a complete determination of their fixed-point on compacta property. The thesis also clarifies certain folklore theorems regarding the Gromov hyperbolic spaces and the dynamics of amenable groups on them. Last but not the least, the thesis studies the connectivity of the Gromov boundary of fine curve graphs, a combinatorial tool employed in the study of the homeomorphism groups of surfaces of finite type
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Ross, Skyler W. "Non-Euclidean Geometry." Fogler Library, University of Maine, 2000. http://www.library.umaine.edu/theses/pdf/RossSW2000.pdf.
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Dario, Douglas Francisco. "Geometrias não euclidianas: elíptica e hiperbólica no ensino médio." Universidade Tecnológica Federal do Paraná, 2014. http://repositorio.utfpr.edu.br/jspui/handle/1/862.
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Este trabalho tem como objetivo colaborar na inserção do ensino das Geometrias Não
Euclidianas no ensino médio. Para tanto, fizemos uma pesquisa bibliográfica sobre o
surgimento de tais Geometrias, em seguida apresentamos uma sequência de
conteúdos para o ensino das Geometrias Elíptica e Hiperbólica, abordando os
principais tópicos elencados pelas Diretrizes Curriculares do Estado do Paraná,
comparando-as sempre que possível com a Geometria Euclidiana. Esclarecemos que
onde citamos Geometria Elíptica, estamos realmente tratando da Geometria da
Superfície Esférica, para que este trabalho fique compatível com as Diretrizes
Curriculares do Estado do Paraná. Apesar de haver algumas proposições e suas
provas, em grande parte do trabalho não há teoria e demonstrações com o rigor
exigido pela matemática, buscamos apenas apresentar os principais conceitos e usar
uma linguagem que possa ser compreendida por qualquer profissional que esteja
disposto a compreender e depois de estudar, ensinar estas geometrias. Em novembro
de 2013, na XVII Semana da Matemática e III Encontro de Ensino de Matemática do
Câmpus de Pato Branco – PR da UTFPR, aplicamos um minicurso com parte deste
conteúdo. Ao final do minicurso aplicamos um questionário sobre o conhecimento
inicial do tema e a atual situação de ensino destas geometrias. Tal questionário visou
identificar o interesse sobre o tema e sobre a real possibilidade de inserção destas
geometrias nas salas de aula, cujos resultados encontram-se no texto.This work aims to contribute in including teaching of Non-Euclidean Geometry in high school. For this, a bibliographic research was made about the appearance of such geometries and introduce content for teaching of Elliptical and Hyperbolic Geometries, addressing the main topics listed by Curriculum Guidelines of Paraná, comparing them with Euclidean Geometry. Clarify that where quoted elliptic geometry, we are really dealing with Surface Spherical Geometry, for that this work be compatible with the Curriculum Guidelines of the State of Paraná. Although there are some propositions and their proofs, in most part of the work there aren´t theoretical studies and statements with all rigors mathematics requires, we seek to show the main concepts and use a language that can be understood by any person who is willing to understand and after studying, teach these geometries in school. In November 2013, during the XVII Semana de Matemática and III Encontro de Ensino de Matemática Câmpus de Pato Branco – PR of UTFPR, a mini-course was applied with part of this content to some participants. At the end of the mini-course a questionnaire was applied inquiring the basic knowledge, the current teaching situation of these geometries and aim to identify the interest in this issue and the real possibility of inclusion in the classrooms, the results can be found in the following work.
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Senger, Steven Iosevich Alex. "Erdős distance problem in the hyperbolic half-plane." Diss., Columbia, Mo. : University of Missouri--Columbia, 2009. http://hdl.handle.net/10355/5341.
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The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on January 14, 2010). Thesis advisor: Dr. Alex Iosevich. Includes bibliographical references.
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Valero, Carlos. "On the geometry and topology of hyperbolic variational symbols." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302354.
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Butler, Joe R. "The Torus Does Not Have a Hyperbolic Structure." Thesis, University of North Texas, 1992. https://digital.library.unt.edu/ark:/67531/metadc500333/.
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Several basic topics from Algebraic Topology, including fundamental group and universal covering space are shown. The hyperbolic plane is defined, including its metric and show what the "straight" lines are in the plane and what the isometries are on the plane. A hyperbolic surface is defined, and shows that the two hole torus is a hyperbolic surface, the hyperbolic plane is a universal cover for any hyperbolic surface, and the quotient space of the universal cover of a surface to the group of automorphisms on the covering space is equivalent to the original surface.
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Praggastis, Brenda L. "Markov partitions for hyperbolic toral automorphisms /." Thesis, Connect to this title online; UW restricted, 1992. http://hdl.handle.net/1773/5773.
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Logan, Deborah F. "A General Theory of Geodesics with Applications to Hyperbolic Geometry." UNF Digital Commons, 1995. http://digitalcommons.unf.edu/etd/101.
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In this thesis, the geometry of curved surfaces is studied using the methods of differential geometry. The introduction of manifolds assists in the study of classical two-dimensional surfaces. To study the geometry of a surface a metric, or way to measure, is needed. By changing the metric on a surface, a new geometric surface can be obtained. On any surface, curves called geodesics play the role of "straight lines" in Euclidean space. These curves minimize distance locally but not necessarily globally. The curvature of a surface at each point p affects the behavior of geodesics and the construction of geometric objects such as circles and triangles. These fundamental ideas of manifolds, geodesics, and curvature are developed and applied to classical surfaces in Euclidean space as well as models of non-Euclidean geometry, specifically, two-dimensional hyperbolic space.
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Jones, Gavin L. "The iteration theory of Blaschke products." Thesis, University of Cambridge, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.308237.
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Haxhi, Karen Kleinschmidt. "The euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs /." View abstract, 1998. http://library.ctstateu.edu/ccsu%5Ftheses/1491.html.
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Thesis (M.S.) -- Central Connecticut State University, 1998.Thesis advisor: Dr. Jeffrey McGowan. "...in partial fulfillment of the requirements for the degree of Master of Science." Includes bibliographical references (leaves [78-79]).
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Masters, Joseph David. "Lengths and homology of hyperbolic 3-manifolds /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.
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Kuhlmann, Sally Malinda. "Geodesic knots in hyperbolic 3 manifolds." Connect to thesis, 2005. http://repository.unimelb.edu.au/10187/916.
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This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds.Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded.
The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially.
We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.
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Venzke, Rupert William Calegari Danny Calegari Danny. "Braid forcing, hyperbolic geometry, and pseudo-Anosov sequences of low entropy /." Diss., Pasadena, Calif. : Caltech, 2008. http://resolver.caltech.edu/CaltechETD:etd-05292008-085545.
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FARACO, Gianluca. "The Geometry Awakens: on the relationship between holonomy and hyperbolic structures." Doctoral thesis, Università degli studi di Ferrara, 2018. http://hdl.handle.net/11392/2487998.
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This thesis examines the relationship between branched hyperbolic structures on surfaces and representations of the fundamental group into $\pslr$. A branched hyperbolic structure on a surface $S$ is a hyperbolic cone-structure such that the angle around any interior cone point is an integer multiple of $2\pi$.\\ Any such structure determines a holonomy representation of the fundamental group $\rho:\pi_1S\longrightarrow \pslr$. We ask, conversely, when a representation of the fundamental group arises as holonomy of a branched hyperbolic structure. We consider only $2-$dimensional hyperbolic structures.\\
In this work, we take into account Mathews's theorems and we improve them. Let $S$ be a closed surface of genus $2$, then we show that any representation $\rho:\pi_1 S\longrightarrow \pslr$ with Euler class $\mathcal{E}(\rho)=\pm 1$ is the holonomy of a branched hyperbolic structure $\sigma$ on $S$. In order to show this, we prove that any such representation sends a simple non-separating curve to an elliptic element. Also, we need to consider separately a special class of representations, namely representations with virtually abelian pairs. This class of representations turns out to problematic, and we need to deal with them in a different way. The same ideas we used in the genus $2$ case can be used in all genus. Using them we may prove that any representation $\rho:\pi_1 S\longrightarrow \pslr$ with Euler class $\mathcal{E}(\rho)=\pm\big(\chi(S)+ 1\big)$ which sends a non-separating simple curve to a non-hyperbolic element, is the holonomy of a branched hyperbolic structure $\sigma$ on $S$ with one cone point of angle $4\pi$.Questa tesi esamina la relazione tra strutture iperboliche ramificate su superfici e rappresentazioni del gruppo fondamentale in $ \ pslr $. Una struttura iperbolica ramificata su una superficie $ S $ è una struttura iperbolica conica tale che l'angolo attorno a qualsiasi punto conico è un multiplo intero di $ 2 \ pi $. \\ Qualsiasi struttura di questo tipo determina una rappresentazione del gruppo fondamentale $ \ rho: \ pi_1S \ longrightarrow \ pslr $ detta olonomia. Ci chiediamo, al contrario, quando una rappresentazione del gruppo fondamentale sorge come olonomia di una struttura iperbolica ramificata. In questo lavoro considereremo solo strutture iperboliche $ 2- $ dimensionali. \\ Sia $ S $ una superficie chiusa di genere $ 2 $, e sia $ \ rho: \ pi_1 S \ longrightarrow \ pslr $ una rappresentazione con classe di Eulero $ \ mathcal {E} (\ rho) = \ pm 1 $. Allora $\rho$ è olonomia di una struttura iperbolica ramificata $ \ sigma $ su $ S $ con un punto conico $4\pi$. Le stesse tecniche usate nel caso del genere $ 2 $ possono essere adoperate anche in genere più alto. In questo modo possiamo provare che qualsiasi rappresentazione $ \ rho: \ pi_1 S \ longrightarrow \ pslr $ con la classe Euler $ \ mathcal {E} (\ rho) = \ pm \ big (\ chi (S) + 1 \ big) $ che applica una curva semplice non separante ad un elemento non iperbolico, è l'olonomia di una struttura iperbolica ramificata $ \ sigma $ su $ S $ con un punto cono di angolo $ 4 \ pi $.
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Valério, José Carlos. "Introdução à geometria hiperbólica." Universidade Federal de Juiz de Fora (UFJF), 2017. https://repositorio.ufjf.br/jspui/handle/ufjf/5405.
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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Na presente dissertação será introduzido o desenvolvimento histórico da Geometria Hiperbólica. Será apresentado o quinto postulado de Euclides, de acordo com o ponto de vista dos Axiomas de Hilbert, correlacionando-os com os resultados da Geometria Neutra. Serão apresentados e provados alguns resultados da Geometria Hiperbólica, no que diz respeito às propriedades das retas paralelas, dos triângulos generalizados e seus critérios de congruência. Por fim, serão discutidas as propriedades que são válidas tanto para a Geometria Euclidiana quanto Hiperbólica, enfatizando que a principal diferença entre elas é o postulado das paralelas.
In the present dissertation we will introduce the historical development of the hyperbolic geometry. We will present Euclid’s fifth postulate from the Hilbert’s axioms point of view and we will correlate them with results of the Neutral Geometry. We will present and prove some results of the Hyperbolic Geometry, regarding the properties of the parallel lines, and the generalized triangles and their congruence criteria. At last, we will discuss the proprieties which are valid in both Euclidean and Hyperbolic Geometry, and we will emphasize that the main difference between them is the parallel postulate.
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Saber, Hicham. "Equivariant Functions for the Möbius Subgroups and Applications." Thèse, Université d'Ottawa / University of Ottawa, 2011. http://hdl.handle.net/10393/20236.
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The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions.
We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions.
In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.
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Marfai, Frank S. "Hyperbolic transformations on cubics in H²." CSUSB ScholarWorks, 2003. https://scholarworks.lib.csusb.edu/etd-project/142.
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The purpose of this thesis is to study the effects of hyperbolic transformations on the cubic that is determined by locus of centroids of the equilateral triangles in H² whose base coincides with the line y=0, and whose common vertex is at the origin. The derivation of the formulas within this work are based on the Poincaré disk model of H², where H² is understood to mean the hyperbolic plane. The thesis explores the properties of both the untransformed cubic (the original locus of centroids) and the transformed cubic (the original cubic taken under a linear fractional transformation).
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Silva, Paul Jerome. "Investigation of the effectiveness of interface constraints in the solution of hyperbolic second-order differential equations." CSUSB ScholarWorks, 2000. https://scholarworks.lib.csusb.edu/etd-project/1953.
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Solutions to differential equations describing the behavior of physical quantities (e.g., displacement, temperature, electric field strength) often only have finite range of validity over a subdomain. Interest beyond the subdomain often arises. As a result, the problem of making the solution compatible across the connecting subdomain interfaces must be dealt with. Four different compatibility methods are examined here for hyperbolic (time varying) second-order differential equations. These methods are used to match two different solutions, one in each subdomain along the connecting interface. The entire domain that is examined here is a unit square in the Cartesian plane. The four compatibility methods examined are: point collocation; optimal least square fit; penalty function; Ritz-Galerkin weak form. Discretized L2 convergence is used to examine and compare the effectiveness of each method.
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Haapalainen, M. (Mikko). "Dielectrophoretic mobility of a spherical particle in 2D hyperbolic quadrupole electrode geometry." Doctoral thesis, Oulun yliopisto, 2013. http://urn.fi/urn:isbn:9789526202648.
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Abstract The dielectric properties of material are of importance in various industrial and scientific applications of measurement technology. These properties are usually measured by microwave sensors, methods that provide an average result of the entire sample volume. However, the need to know the dielectric properties of single particles has increased in various segments. Dielectrophoresis (DEP) as a method provides a way to determine dielectric properties, such as permittivity and conductivity, of single particles with good precision by using a sufficiently simple system. In this thesis the DEP platform consists of four electrodes with hyperbolic edge geometry. Geometry produces a linear electric field gradient which leads to a constant DEP force in the active region and ensures fairly straightforward determination of particle properties by means of their DEP mobility. Particle mobility is dependent on particle size, the slope of the electric field gradient, the conductivity of the carrier fluid and the frequency of the electric field. For this thesis has been studied particle characterization on a 2D platform, particle 3D behavior, simultaneous multiphenomena observation and the applicability of transparent indium-tin-oxide (ITO) electrode material to DEP platforms. Experiments were made with polystyrene particles, using carrier fluids of varying conductivity. Electrode transparency enabled holographic 3D imaging and thus also the observation of particle behavior in the depth direction. This 3D imaging revealed the restricted working distance from the electrode plane, as well as the mobility of particles in the depth dimension under the DEP force. It was also observed that there was only a marginal difference between 3D mobility and 2D mobility inside the active region. The dielectric properties of the polystyrene particles were determined and found to differ from those of solid polystyrene, and thus the properties of the carrier fluid have a distinctive effect on particle properties. The measurement achieved good precision in a single particle measurement and thus the total particle consistency could be very low. The main restrictions of the DEP method are the limited working distance and restricted range of particle sizes: based on the results of this thesis, with a fixed size platform the particle size may not vary by more than tenfoldTiivistelmä Materiaalin dielektriset ominaisuudet ovat merkittäviä monissa teollisissa sekä tieteellisissä mittausteknisissä sovelluksissa. Yleensä nämä ominaisuudet mitataan mikroaaltosensoreilla, joilla mitataan näytetilavuuden keskiarvo. Tarve tietää yksittäisen partikkelin dielektriset ominaisuudet on kuitenkin lisääntynyt useilla segmenteillä. Dielektroforeesi (DEP) on menetelmä, jolla voidaan toistettavasti mitata yksittäisen partikkelin permittiivisyys ja johtavuus suhteellisen yksinkertaisella mittaussysteemillä. Tässä työssä DEP-alusta koostuu neljästä elektrodista, joilla on hyperbelin muotoinen elektrodin reunan muoto. Geometria tuottaa lineaarisen sähkökentän gradientin, mikä johtaa vakio DEP-voimaan alustan aktiivisella alueella. Niinpä partikkelin ominaisuuksien määrittely niiden liikkuvuuden perusteellä on suoraviivaista. Liikkuvuus on riippuvainen partikkelin koosta, sähkökentän gradientin kulmakertoimesta, kantajaliuoksen johtavuudesta sekä käytetyn sähkökentän taajuudesta. Työssä on tutkittu partikkelin karakterisointia 2D-alustalla, partikkelin 3D-käyttäytymistä, samanaikaista moni-ilmiötarkkailua sekä indium-tina-oksidi (ITO) elektrodimateriaalin soveltuvuutta DEP-alustoihin. Kokeet tehtiin polystyreenipartikkeleilla käyttäen eri johtavuuksisia kaliumkloridi-kantajaliuoksia. Elektrodien läpinäkyvyys mahdollisti holografisen 3D-kuvantamisen, jonka avulla havaittiin alustan rajallinen toimintaetäisyys sekä partikkelin syvyyssuuntainen liike DEP-voiman vaikutuksesta. Kuitenkin 3D- ja 2D-liikkeen välillä havaittiin vain vähäinen ero. Määritellyt partikkelin dielektriset ominaisuudet eroavat kiinteän polystyreenin arvoista, jolloin kantajaliuoksen ominaisuuksilla on merkittävä vaikutus määriteltyihin partikkelin ominaisuuksiin. Mittauksilla saavutettiin hyvä toistettavuus jopa yksittäisen partikkelin mittauksista, niinpä näytteen partikkelipitoisuus voi olla erittäin pieni. DEP-menetelmän merkittävimpiä rajoitteita ovat rajallinen toimintaetäisyys sekä rajallinen partikkelikoon variaatio, mikä voi olla yhdellä DEP-alustalla noin kymmenkertainen
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ROSMONDI, DANIELE. "Earthquakes on hyperbolic surfaces with geodesic boundary and Anti de Sitter geometry." Doctoral thesis, Università degli studi di Pavia, 2017. http://hdl.handle.net/11571/1203304.
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Rossini, Marcela Aparecida Penteado. "Um estudo sobre o uso de régua, compasso e um software de geometria dinâmica no ensino da geometria hiperbólica /." Rio Claro : [s.n.], 2010. http://hdl.handle.net/11449/91137.
Full textAbstract:
Orientador: Claudemir MurariBanca: Ruy Madsen Barbosa
Banca: Geraldo Perez
Resumo: O principal objetivo deste trabalho foi contribuir para o ensino e aprendizagem da geometria hiperbólica, apresentando uma proposta que visa à introdução do estudo dessa geometria, utilizando o software Cabri - Géomètre II (menu hiperbólico) e, também, a régua e o compasso na abordagem dos conceitos fundamentais. Procedemos desta maneira por entendermos que estes recursos integrados podem cooperar para uma melhor compreensão e assimilação das noções apresentadas. Esta pesquisa tem abordagem do tipo qualitativa e foi desenvolvida seguindo a proposta metodológica de Romberg, e a coleta de dados se deu por meio de observações, anotações e fotos. A metodologia de resolução de problemas foi utilizada na elaboração das atividades, as quais foram aplicadas a alunos de um curso de graduação em engenharia elétrica. Os dados coletados foram analisados qualitativamente, buscando compreender como tais instrumentos educacionais associados podem auxiliar no processo ensino-aprendizagem desta geometria não-euclidiana. Ressaltamos que na evolução desta proposta foi possível reforçar o entendimento de conceitos da geometria euclidiana que são usados nas construções hiperbólicas
Abstract: The ultimate goal of this essay is to contribute to the teaching and learning process of the hyperbolic geometry,introducing a proposal that aims for the presentation of the study of this geometry,making use of the software entitled Geomere Cabri ll ( hyperbolic menu ) and , also ,the ruler and the compasses in the approach of essential concepts. We followed this way of working as we understand that such resources when integrated, may cooperate in a better understanding and assimilation of those presented notions.This research has a qualitative kind of approach and was developed following the methodological proposal of Romberg, and we collect data by watching ,writing down notes and taking pictures.The methodology of problem solutions was deployed in the activities elaboration which were applied to students in an Electric Engineering degree course.The collected data were analyzed taking into account the quality, trying to understand how such educational tools together may help in the teaching- learningprocess of these no - Euclidian geometry concepts which are used in the hyperbolic constructions
Mestre
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Weed, Michael. "The anatomy of hyperbolic trajectories in the Gulf of Mexico." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file 2.59 Mb., 169 p, 2006. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:1432294.
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Hidalgo, Joshua L. "A KLEINIAN APPROACH TO FUNDAMENTAL REGIONS." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/35.
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This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be linear fractional transformations when in $U^2$ and $B^2$. We will then study a bit of topology, specifically the topological groups relevant to the group of isometries of hyperbolic $n$-space, $I(H^n)$. Finally we will combine what we know about hyperbolic $n$-space and topological groups in order to study fundamental regions, fundamental domains, Dirichlet domains, and quotient spaces. Using examples in $U^2$, we will then illustrate how useful fundamental domains are when it comes to visualizing the geometry of quotient spaces.
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Arcari, Inedio. "Um texto de geometria hiperbólica." [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307016.
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Orientador: Edson AgustiniDissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação
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Resumo: A presente dissertação é um texto introdutório de Geometria Hiperbólica com alguns resultados e comentários de Geometria Elíptica. Nossa intenção foi compilar um material que possa ser utilizado em cursos introdutórios de Geometria Hiperbólica tanto em nível de licenciatura quanto de bacharelado. Para tornar o texto mais acessível, notas históricas sobre a bela página do desenvolvimento das Geometrias Não Euclidianas foram introduzidas logo no primeiro capítulo. Procuramos ilustrar fartamente o texto com figuras dentre as quais várias que foram baseadas no Modelo Euclidiano do Disco de Poincaré para a Geometria Hiperbólica. Atualmente, o estudo de Geometria Hiperbólica tem sido bastante facilitado pelo uso de softwares de geometria dinâmica, como o Cabri-Géometre, GeoGebra e NonEuclid, sendo esses dois últimos softwares livres
Abstract: The present work is an introductory text of Hyperbolic Geometry with some results and comments of Elliptic eometry. Our aim in this work were to compile a material that can be used as introduction to Hyperbolic Geometry inundergraduated courses. In the first chapter we introduced historical notes about the beautiful development of the Non Euclid Geometries in order to turn the text more interesting and accesible. We illustrated the text with many figures which were done on the Euclidean Model of the Poincaré' s Disk for the Hyperbolic Geometry. In this way, the study of Hyperbolic Geometry has been softened by the use of softwares of dynamic geometry, like Cabri-Geométre and the freeware softwares GeoGebra and NonEuclid
Mestrado
Mestre em Matemática
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Kume, Alfred. "On the Frechet means in simplex shape spaces." Thesis, University of Nottingham, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368237.
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Franco, Felipe de Aguilar. "On spaces of special elliptic n-gons." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-22032019-081425/.
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We study relations between special elliptic isometries in the complex hyperbolic plane. A special elliptic isometry can be seen as a rotation around a fixed axis (a complex geodesic). Such an isometry is determined by specifying a nonisotropic point p (the polar point to the fixed axis) and a unitary complex number a, the angle of the isometry. Any relation between special elliptic isometries with rational angles gives rise to a representation H(k1;:::;kn) → PU(2;1), where H(k1;:::;kn) : = ⟨ r1; : : : ; rn ∣ rn : : : r1> = 1; rkii = 1 ⟩ and PU(2;1) stands for the group of orientation-preserving isometries of the complex hyperbolic plane. We denote by Rpα the special elliptic isometry determined by the nonisotropic point p and by the unitary complex number α. Relations of the form Rpnαn : : :Rp1α1 = 1 in PU(2;1), called special elliptic n-gons, can be modified by short relations known as bendings: given a product RqβRpα, there exists a one-parameter subgroup B : R → SU(2;1) such that B(s) is in the centralizer of Rqβ Rpα and RB(s)qβRB(s)pα = RqβRB(s)pα for every s ∈ R. Then, for each i = 1,...,n-1, we can change Rpi+1αi+1Rpiαi by RB(s)pi+1αi+1RB(s)piαi obtaining a new n-gon. We prove that the generic part of the space of pentagons with fixed angles and signs of points is connected by means of bendings. Furthermore, we describe certain length 4 relations, called f -bendings, and prove that the space of pentagons with fixed product of angles is connected by means of bendings and f -bendings.Neste trabalho, estudamos relações entre isometrias elípticas especiais no plano hiperbólico complexo. Uma isometria elíptica especial pode ser vista como uma rotação em torno de um eixo fixo (uma geodésica complexa). Tal isometria é determinada especificando-se um ponto não-isotrópico p (o ponto polar do eixo fixo) bem como um número complexo unitário a (o ângulo da isometria). Qualquer relação entre isometrias elípticas especiais com ângulos racionais dá origem a uma representação H(k1;:::;kn) → PU(2;1), onde H(k1;:::;kn) : = ⟨ r1; : : : ; rn ∣ rn : : : r1 = 1; rkii = 1 ⟩ e PU(2;1) é o grupo de isometrias que preservam a orientação do plano hiperbólico complexo. Denotamos por Rpα a isometria elíptica especial determinada pelo ponto não-isotrópico p e pelo complexo unitário α. Relações da forma Rpnαn : : :Rp1α1 = 1 em PU(2;1), chamadas n-ágonos elípticos especiais, podem ser modificadas a partir de relações curtas conhecidas como bendings: dado um produto RqβRpα, existe um subgrupo uniparamétrico B : R → SU(2;1) tal que B(s) está no centralizador de RqβRpα e RB(s)qβRB(s)pα = RqβRpα para todo s ∈ R. Assim, para cada i = 1; : : : ;n-1, podemos mudar Rpi+1α+1Rpiαi por RB(s)pi+1α+1RB(s)piα+1RB(s)piαi obtendo um novo n-ágono. Provamos que a parte genérica do espaço de pentágonos com ângulos e sinais de pontos fixados é conexa por meio de bendings. Além disso, descrevemos certas relações de comprimento 4, os f -bendings, e provamos que o espaço de pentágonos com produto de ângulos fixado é conexo por meio de bendings e f -bendings.
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Garza, César. "Examples of hyperbolic knots with distance 3 toroidal surgeries in S³." To access this resource online via ProQuest Dissertations and Theses @ UTEP, 2009. http://0-proquest.umi.com.lib.utep.edu/login?COPT=REJTPTU0YmImSU5UPTAmVkVSPTI=&clientId=2515.
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Burton, Stephan Daniel. "Unknotting Tunnels of Hyperbolic Tunnel Number n Manifolds." BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3307.
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Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n−2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n−1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the generalization of Adam's conjecture to tunnel number n ≥ 2 manifolds may be false.
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Sisto, Alessandro. "Geometric and probabilistic aspects of groups with hyperbolic features." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:bcf456c4-eef0-4fe8-bb7d-8b15f9cf7b18.
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The main objects of interest in this thesis are relatively hyperbolic groups. We will study some of their geometric properties, and we will be especially concerned with geometric properties of their boundaries, like linear connectedness, avoidability of parabolic points, etc. Exploiting such properties will allow us to construct, under suitable hypotheses, quasi-isometric embeddings of hyperbolic planes into relatively hyperbolic groups and quasi-isometric embeddings of relatively hyperbolic groups into products of trees. Both results have applications to fundamental groups of 3-manifolds. We will also study probabilistic properties of relatively hyperbolic groups and of groups containing ``hyperbolic directions'' despite not being relatively hyperbolic, like mapping class groups, Out(Fn), CAT(0) groups and subgroups of the above. In particular, we will show that the elements that generate the ``hyperbolic directions'' (hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, fully irreducible elements in Out(Fn) and rank one elements in CAT(0) groups) are generic in the corresponding groups (provided at least one exists, in the case of CAT(0) groups, or of proper subgroups). We also study how far a random path can stray from a geodesic in the context of relatively hyperbolic groups and mapping class groups, but also of groups acting on a relatively hyperbolic space. We will apply this, for example, to show properties of random triangles.
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Anaya, Bob. "Fuchsian Groups." CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/838.
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Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.
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Pilla, Eliane Cristina Geroli. "Construções de constelações de sinais geometricamente uniformes hiperbólicas." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/259790.
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Orientador: Reginaldo Palazzo JúniorDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
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Resumo: O presente trabalho tem como meta principal construir constelações de sinais geometricamente uniformes no plano hiperbólico, visando considerá-las como alfabeto para geração de códigos de espaço de sinais, em particular os códigos de classes laterais generalizados. Para estabelecer estas constelações foi escolhido um conjunto de sinais geometricamente uniforme, constituído pelos centros dos octógonos da tesselação {8, 8}. Depois foi obtido um rotulamento para os elementos do grupo gerador dos conjuntos de sinais geometricamente uniformes em cada classe lateral. Finalmente, a partir do isomorfismo rótulo obtivemos um rotulamento isométrico para os elementos do conjunto de sinais
Abstract: Our goal in this work is to construct hyperbolic geometrically uniform signal constellations (more specifically g-torus) that are able to act as alphabets for ge neration of codes. To obtain these constellations we choose geometrically uniform signal sets consisting of the centers of the p-gons of tessellations of type {p, q}. From these constellations we obtain labelings for the elements of the generator group of the geometrically uniform signal sets in each coset. Finally, by the label isomorphism we obtain an isometric labeling for the elements of the signal set
Mestrado
Telecomunicações e Telemática
Mestre em Engenharia Elétrica
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Persson, Anna. "Grundläggande hyperbolisk geometri." Thesis, Karlstad University, Faculty of Technology and Science, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-211.
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I denna uppsats presenteras grundläggande delar av hyperbolisk geometri. Uppsatsen är indelad i två kapitel. I första kapitlet studeras Möbiusavbildningar på Riemannsfären. Andra kapitlet presenterar modellen av hyperbolisk geometri i övre halvplanet H, skapad av Poincaré på 1880-talet.
Huvudresultatet i uppsatsen är Gauss – Bonnét´s sats för hyperboliska trianglar.
In this thesis we present fundamental concepts in hyperbolic geometry. The thesis is divided into two chapters. In the first chapter we study Möbiustransformations on the Riemann sphere. The second part of the thesis deal with hyperbolic geometry in the upper half-plane. This model of hyperbolic geometry was created by Poincaré in 1880.
The main result of the thesis is Gauss – Bonnét´s theorem for hyperbolic triangles.
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Galli, Daniele. "The Selberg Zeta Function: A golden thread through hyperbolic geometry, dynamics and number theory." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18781/.
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This thesis deals with the profound relationship that exists between the dynamics on surfaces of negative curvature, some spectral aspects of hyperbolic geometry, and the ergodic properties of continued fractions. One of the most elegant results to reveal the connections between the geodesic flow on a surface of constant negative curvature and the spectrum of the Laplace-Beltrami operator is the celebrated trace formula of Selberg. From a standpoint which is not that of this thesis, one could say that any such trace formula represents one of the most direct connections between observations of classical dynamics and observations of quantum dynamics. At any rate, the main mathematical tool behind the original trace formula is the Selberg zeta function. The purpose of the thesis is to review certain papers of D. H. Mayer, and more recently of C.Bonanno and S. Isola, on certain representations of the Selberg zeta function for surfaces of constant negative curvature based on families of transfer operators for the Gauss and Farey maps, which are the dynamical systems associated to the continued-fraction expansion of real numbers. In particular, we extend some results of Bonanno and Isola to subgroups of the full modular group PSL(2;Z);
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Biswas, Debapriya. "Geometry of elliptic, parabolic and hyperbolic homogeneous spaces using Clifford algebras and group representations." Thesis, University of Leeds, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.432297.
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Strzheletska, Elena. "The Euler Line in non-Euclidean geometry." CSUSB ScholarWorks, 2003. https://scholarworks.lib.csusb.edu/etd-project/2443.
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The main purpose of this thesis is to explore the conditions of the existence and properties of the Euler line of a triangle in the hyperbolic plane. Poincaré's conformal disk model and Hermitian matrices were used in the analysis.ʹ
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Rossini, Marcela Aparecida Penteado [UNESP]. "Um estudo sobre o uso de régua, compasso e um software de geometria dinâmica no ensino da geometria hiperbólica." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/91137.
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rossini_map_me_rcla.pdf: 3173837 bytes, checksum: 262f12484461eccfd7aff81dfec3e0c2 (MD5)O principal objetivo deste trabalho foi contribuir para o ensino e aprendizagem da geometria hiperbólica, apresentando uma proposta que visa à introdução do estudo dessa geometria, utilizando o software Cabri - Géomètre II (menu hiperbólico) e, também, a régua e o compasso na abordagem dos conceitos fundamentais. Procedemos desta maneira por entendermos que estes recursos integrados podem cooperar para uma melhor compreensão e assimilação das noções apresentadas. Esta pesquisa tem abordagem do tipo qualitativa e foi desenvolvida seguindo a proposta metodológica de Romberg, e a coleta de dados se deu por meio de observações, anotações e fotos. A metodologia de resolução de problemas foi utilizada na elaboração das atividades, as quais foram aplicadas a alunos de um curso de graduação em engenharia elétrica. Os dados coletados foram analisados qualitativamente, buscando compreender como tais instrumentos educacionais associados podem auxiliar no processo ensino-aprendizagem desta geometria não-euclidiana. Ressaltamos que na evolução desta proposta foi possível reforçar o entendimento de conceitos da geometria euclidiana que são usados nas construções hiperbólicas
The ultimate goal of this essay is to contribute to the teaching and learning process of the hyperbolic geometry,introducing a proposal that aims for the presentation of the study of this geometry,making use of the software entitled Geomere Cabri ll ( hyperbolic menu ) and , also ,the ruler and the compasses in the approach of essential concepts. We followed this way of working as we understand that such resources when integrated, may cooperate in a better understanding and assimilation of those presented notions.This research has a qualitative kind of approach and was developed following the methodological proposal of Romberg, and we collect data by watching ,writing down notes and taking pictures.The methodology of problem solutions was deployed in the activities elaboration which were applied to students in an Electric Engineering degree course.The collected data were analyzed taking into account the quality, trying to understand how such educational tools together may help in the teaching- learningprocess of these no - Euclidian geometry concepts which are used in the hyperbolic constructions
50
Genevois, Anthony. "Cubical-like geometry of quasi-median graphs and applications to geometric group theory." Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0569/document.
Full textAbstract:
La classe des graphes quasi-médians est une généralisation des graphes médians, ou de manière équivalente, des complexes cubiques CAT(0). L'objectif de cette thèse est d'introduire ces graphes dans le monde de la théorie géométrique des groupes. Dans un premier temps, nous étendons la notion d'hyperplan définie dans les complexes cubiques CAT(0), et nous montrons que la géométrie d'un graphe quasi-médian se réduit essentiellement à la combinatoire de ses hyperplans. Dans la deuxième partie de notre texte, qui est le cœur de la thèse, nous exploitons la structure particulière des hyperplans pour démontrer des résultats de combinaison. L'idée principale est que si un groupe agit d'une bonne manière sur un graphe quasi-médian de sorte que les stabilisateurs de cliques satisfont une certaine propriété P de courbure négative ou nulle, alors le groupe tout entier doit satisfaire P également. Les propriétés que nous considérons incluent : l'hyperbolicité (éventuellement relative), les compressions lp (équivariantes), la géométrie CAT(0) et la géométrie cubique. Finalement, la troisième et dernière partie de la thèse est consacrée à l'application des critères généraux démontrés précédemment à certaines classes de groupes particulières, incluant les produits graphés, les groupes de diagrammes introduits par Guba et Sapir, certains produits en couronne, et certains graphes de groupes. Les produits graphés constituent notre application la plus naturelle, où le lien entre le groupe et son graphe quasi-médian associé est particulièrement fort et explicite; en particulier, nous sommes capables de déterminer précisément quand un produit graphé est relativement hyperboliqueThe class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its hyperplanes. In the second part, we exploit the specific structure of the hyperplanes to state combination results. The main idea is that if a group acts in a suitable way on a quasi-median graph so that clique-stabilisers satisfy some non-positively curved property P, then the whole group must satisfy P as well. The properties we are interested in are mainly (relative) hyperbolicity, (equivariant) lp-compressions, CAT(0)-ness and cubicality. In the third part, we apply our general criteria to several classes of groups, including graph products, Guba and Sapir's diagram products, some wreath products, and some graphs of groups. Graph products are our most natural examples, where the link between the group and its quasi-median graph is particularly strong and explicit; in particular, we are able to determine precisely when a graph product is relatively hyperbolic