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130 relations: Abelian group, Acta Mathematica, Advances in Mathematics, Alexander Lubotzky, Algebraic geometry, Algebraic group, Algebraic topology, Amenable group, André Haefliger, Annals of Mathematics, Archiv der Mathematik, Arithmetic group, Artin group, Automatic group, Bass–Serre theory, Baum–Connes conjecture, Baumslag–Solitar group, Braid group, Brian Bowditch, Building (mathematics), Burnside problem, C*-algebra, CAT(k) group, CAT(k) space, Cayley graph, Combinatorial group theory, Communications on Pure and Applied Mathematics, Computational complexity theory, Computational group theory, Convergence group, Coxeter group, Cyclic group, Differential geometry, Discrete group, Dodecahedron, Dynamical system, Egbert van Kampen, Eliyahu Rips, European Congress of Mathematics, Felix Klein, Finite subdivision rule, Finitely generated group, Free group, Free product, Fuchsian group, Generic-case complexity, Geometric analysis, Geometrization conjecture, Geometry, Geometry & Topology, ..., Georges de Rham, Graph of groups, Grigorchuk group, Gromov boundary, Gromov's theorem on groups of polynomial growth, Group action, Group cohomology, Growth rate (group theory), Hilbert space, Homeomorphism, Hyperbolic geometry, Hyperbolic group, Icosahedral symmetry, Icosian calculus, Integer, Inventiones Mathematicae, Iterated monodromy group, J. H. C. Whitehead, Jakob Nielsen (mathematician), John Stillwell, Journal of Algebra, Journal of the American Mathematical Society, K-theory, Karen Vogtmann, Kazhdan's property (T), Kleinian group, Kurt Reidemeister, Lattice (discrete subgroup), Lie group, Low-dimensional topology, Mapping class group, Mathematical logic, Mathematics, Max Dehn, Metric space, Mikhail Leonidovich Gromov, Mladen Bestvina, Morse theory, Mostow rigidity theorem, Nielsen transformation, Nilpotent group, Novikov conjecture, Otto Schreier, Out(Fn), Outer automorphism group, Pacific Journal of Mathematics, Paul Schupp, Ping-pong lemma, Poisson boundary, Presentation of a group, Probability theory, Quasi-isometry, Quotient group, Random walk, Real tree, Relatively hyperbolic group, Richard Schwartz, Roger Lyndon, Saharon Shelah, Small cancellation theory, Stéphane Mallarmé, Subgroup growth, Symmetric group, Thompson groups, Tietze transformations, Topological dynamics, Topology, Topology (journal), Train track (mathematics), Tree (graph theory), Ultralimit, Van Kampen diagram, Wallpaper group, Walther von Dyck, Wilhelm Magnus, William Rowan Hamilton, William Thurston, Word metric, Word problem for groups, Zlil Sela. Expand index (80 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Acta Mathematica
Acta Mathematica is a peer-reviewed open-access scientific journal covering research in all fields of mathematics.
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Advances in Mathematics
Advances in Mathematics is a mathematics journal publishing research on pure mathematics.
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Alexander Lubotzky
Professor Alexander Lubotzky (אלכסנדר לובוצקי, born 28 June 1956) is an Israeli mathematician and former politician who is currently a professor at the Hebrew University of Jerusalem and an adjunct professor at Yale University.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic group
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
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Amenable group
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.
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André Haefliger
André Haefliger (born 22 May 1929) is a Swiss mathematician who works primarily on topology.
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Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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Archiv der Mathematik
Archiv der Mathematik is a peer-reviewed mathematics journal published by Springer, established in 1948.
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Arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z).
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Artin group
In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form \begin \Big\langle x_1,x_2,\ldots,x_n \Big| \langle x_1, x_2 \rangle^ &.
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Automatic group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata.
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Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees.
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Baum–Connes conjecture
In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group.
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Baumslag–Solitar group
In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases.
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Braid group
In mathematics, the braid group on strands (denoted), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry.
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Brian Bowditch
Brian Hayward Bowditch (born 1961 Bowditch's personal information page at the University of Warwick) is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology.
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Building (mathematics)
In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.
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Burnside problem
The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group.
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C*-algebra
C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.
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CAT(k) group
In mathematics, a CAT(k) group is a group that acts discretely, cocompactly and isometrically on a CAT(''k'') space.
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CAT(k) space
In mathematics, a \mathbf space, where k is a real number, is a specific type of metric space.
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Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group.
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Combinatorial group theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations.
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Communications on Pure and Applied Mathematics
Communications on Pure and Applied Mathematics is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences.
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Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.
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Computational group theory
In mathematics, computational group theory is the study of groups by means of computers.
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Convergence group
In mathematics, a convergence group or a discrete convergence group is a group \Gamma acting by homeomorphisms on a compact metrizable space M in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary \mathbb S^2 of the hyperbolic 3-space \mathbb H^3.
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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Discrete group
In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology.
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Dodecahedron
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.
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Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.
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Egbert van Kampen
Egbert Rudolf van Kampen (28 May 1908, Berchem, Belgium – 11 February 1942, Baltimore, Maryland) was a Dutch mathematician.
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Eliyahu Rips
Eliyahu Rips, also Ilya Rips (אליהו ריפס; Илья Рипс; Iļja Ripss; born 12 December 1948) is an Israeli mathematician of Latvian origin known for his research in geometric group theory.
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European Congress of Mathematics
The European Congress of Mathematics (ECM) is an international congress of the mathematics community, held every four years.
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Felix Klein
Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.
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Finite subdivision rule
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces.
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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 the finite set S and of inverses of such elements.
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Free group
In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st.
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Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties.
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Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,'''R''').
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Generic-case complexity
Generic-case complexity is a subfield of computational complexity theory that studies the complexity of computational problems on "most inputs".
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Geometric analysis
Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations.
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Geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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Geometry & Topology
Geometry & Topology is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications.
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Georges de Rham
Georges de Rham (10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.
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Graph of groups
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups.
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Grigorchuk group
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth.
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Gromov boundary
In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space.
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Gromov's theorem on groups of polynomial growth
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.
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Growth rate (group theory)
In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.
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Hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry.
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Icosahedral symmetry
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.
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Icosian calculus
The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Inventiones Mathematicae
Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.
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Iterated monodromy group
In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering.
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J. H. C. Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as Henry, was a British mathematician and was one of the founders of homotopy theory.
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Jakob Nielsen (mathematician)
Jakob Nielsen (15 October 1890 in Mjels, Als – 3 August 1959 in Helsingør) was a Danish mathematician known for his work on automorphisms of surfaces.
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John Stillwell
John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University.
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Journal of Algebra
Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra.
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Journal of the American Mathematical Society
The Journal of the American Mathematical Society (JAMS), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society.
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K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
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Karen Vogtmann
Karen Vogtmann (born July 13, 1949 in Pittsburg, California Notices of the American Mathematical Society. September 2002, Volume 49, Issue 8, pp. 970–981) is an American mathematician working primarily in the area of geometric group theory.
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Kazhdan's property (T)
In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology.
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Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL(2, '''C''').
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Kurt Reidemeister
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
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Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions.
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Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space.
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Mathematical logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German-born American mathematician and student of David Hilbert.
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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Mikhail Leonidovich Gromov
Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943), is a French-Russian mathematician known for work in geometry, analysis and group theory.
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Mladen Bestvina
Mladen Bestvina (born 1959) is a Croatian-American mathematician working in the area of geometric group theory.
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Morse theory
"Morse function" redirects here.
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Mostow rigidity theorem
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique.
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Nielsen transformation
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups,.
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Nilpotent group
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.
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Novikov conjecture
The Novikov conjecture is one of the most important unsolved problems in topology.
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Otto Schreier
Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was an Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups.
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Out(Fn)
In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators.
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Outer automorphism group
In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.
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Pacific Journal of Mathematics
The Pacific Journal of Mathematics (ISSN 0030-8730) is a mathematics research journal supported by a number of American, Asian and Australian universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation.
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Paul Schupp
Paul Eugene Schupp (born March 12, 1937) is a Professor Emeritus of Mathematics at the University of Illinois at Urbana Champaign.
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Ping-pong lemma
In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.
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Poisson boundary
In mathematics, the Poisson boundary is a measure space associated to a random walk.
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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Probability theory
Probability theory is the branch of mathematics concerned with probability.
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Quasi-isometry
In mathematics, quasi-isometry is an equivalence relation on metric spaces that ignores their small-scale details in favor of their coarse structure.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Random walk
A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
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Real tree
In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees.
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Relatively hyperbolic group
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group.
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Richard Schwartz
Richard Evan Schwartz (born August 11, 1966) is an American mathematician notable for his contributions to geometric group theory and to an area of mathematics known as billiards.
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Roger Lyndon
Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.
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Saharon Shelah
Saharon Shelah (שהרן שלח) is an Israeli mathematician.
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Small cancellation theory
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other.
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Stéphane Mallarmé
Stéphane Mallarmé (18 March 1842 – 9 September 1898), whose real name was Étienne Mallarmé, was a French poet and critic.
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Subgroup growth
In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Thompson groups
In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F \subseteq T \subseteq V, which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to von Neumann conjecture.
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Tietze transformations
In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group.
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Topological dynamics
In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Topology (journal)
Topology was a peer-reviewed mathematical journal covering topology and geometry.
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Train track (mathematics)
In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions.
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Tree (graph theory)
In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path.
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Ultralimit
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces Xn a limiting metric space.
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Van Kampen diagram
In the mathematical area of geometric group theory, a van Kampen diagram (sometimes also called a Lyndon–van Kampen diagram) is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group.
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Wallpaper group
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern.
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Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 in Munich – 5 November 1934 in Munich), born Dyck and later ennobled, was a German mathematician.
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Wilhelm Magnus
Wilhelm Magnus (February 5, 1907, Berlin, Germany – October 15, 1990, New Rochelle, NY) was a German American mathematician.
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William Rowan Hamilton
Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.
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William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.
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Word metric
In group theory, a branch of mathematics, a word metric on a group G is a way to measure distance between any two elements of G. As the name suggests, the word metric is a metric on G, assigning to any two elements g, h of G a distance d(g,h) that measures how efficiently their difference g^ h can be expressed as a word whose letters come from a generating set for the group.
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Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element.
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Zlil Sela
Zlil Sela is an Israeli mathematician working in the area of geometric group theory.
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References
[1] https://en.wikipedia.org/wiki/Geometric_group_theory
