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Density Theorem For Kleinian Groups
In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by and , states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. History suggested the Bers density conjecture, that singly degenerate Kleinian surface groups are on the boundary of a Bers slice. This was proved by for Kleinian surface groups with no parabolic elements. A more general version of Bers's conjecture due to Sullivan and Thurston in the late 1970s and early 1980s states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. proved this for freely indecomposable Kleinian groups without parabolic elements. The density conjecture was finally proved using the tameness theorem and the ending lamination theorem In hyperbolic geometry, the ending lamination theorem, originally conjectured by , states that hyperbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex number, complex matrix (mathematics), matrices of determinant 1 by their center (group theory), center, which consists of the identity matrix and its product by . has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball in . The group of Möbius transformation, Möbius transformations is also related as the non-orientation-preserving isometry group of , . So, a Kleinian group can be regarded as a discrete subgroup group action, acting on one of these spaces. History The theory of general Kleinian groups was founded by and , who named them after Felix Klein. The special case of Schottky groups had been studied a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipman Bers
Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also known for his work in human rights activism.. Biography Bers was born in Riga, then under the rule of the Russian Czars, and spent several years as a child in Saint Petersburg; his family returned to Riga in approximately 1919, by which time it was part of independent Latvia. In Riga, his mother was the principal of a Jewish elementary school, and his father became the principal of a Jewish high school, both of which Bers attended, with an interlude in Berlin while his mother, by then separated from his father, attended the Berlin Psychoanalytic Institute. After high school, Bers studied at the University of Zurich for a year, but had to return to Riga again because of the difficulty of transferring money from Latvia in the international fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University. Sullivan was awarded the Wolf Prize in Mathematics in 2010 and the Abel Prize in 2022. Early life and education Sullivan was born in Port Huron, Michigan, on February 12, 1941.. His family moved to Houston soon afterwards. He entered Rice University to study chemical engineering but switched his major to mathematics in his second year after encountering a particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words: He received his Bachelor of Arts degree from Rice in 1963. He obtained his Doctor of Philosophy from Princeton University in 1966 with his thesis, ''Triangulating h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, he r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A subgroup of a finitely generated group need not be finitely generated. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrically Finite Kleinian Group
In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite groups. Geometrically finite polyhedra A convex polyhedron ''C'' in hyperbolic space is called geometrically finite if its closure in the conformal compactification of hyperbolic space has the following property: *For each point ''x'' in , there is a neighborhood ''U'' of ''x'' such that all faces of meeting ''U'' also pass through ''x'' . For example, every polyhedron with a finite number of faces is geometrically finite. In hyperbolic space of dimension at most 2, every geometrically finite polyhedron has a finite number of sides, but there are geometrically finite polyhedra in dimensions 3 and above with infinitely many sides. For example, in Euclidean space R''n'' of dimension ''n''≥2 there is a polyhedron ''P'' with an infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bers Slice
In the mathematical theory of Kleinian groups, Bers slices and Maskit slices, named after Lipman Bers and Bernard Maskit, are certain slices through the moduli space of Kleinian groups. Bers slices For a quasi-Fuchsian group, the limit set is a Jordan curve whose complement has two components. The quotient of each of these components by the groups is a Riemann surface, so we get a map from marked quasi-Fuchsian groups to pairs of Riemann surfaces, and hence to a product of two copies of Teichmüller space. A Bers slice is a subset of the moduli space of quasi-Fuchsian groups for which one of the two components of this map is a constant function to a single point in its copy of Teichmüller space. The Bers slice gives an embedding of Teichmüller space into the moduli space of quasi-Fuchsian groups, called the Bers embedding, and the closure of its image is a compactification (mathematics), compactification of Teichmüller space called the Bers compactification. Maskit slices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tameness Theorem
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure conjecture. History Topological tameness may be viewed as a property of the ends of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, that is, for surfaces. However, as the example of Alexander horned sphere shows, there are wild embeddings among 3-manifolds, so this property is not automatic. The conjecture was raised in the form of a question by Albert Marden, who proved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ending Lamination Theorem
In hyperbolic geometry, the ending lamination theorem, originally conjectured by , states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic lamination Lamination is the technique/process of manufacturing a material in multiple layers, so that the composite material achieves improved strength, stability, sound insulation, appearance, or other properties from the use of the differing materia ...s on some surfaces in the boundary of the manifold. The ending lamination theorem is a generalization of the Mostow rigidity theorem to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals".. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics". Publication history The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees. Poincaré episode The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
