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49 relations: Algebraic geometry, Annulus (mathematics), Arithmetic group, Braid group, Clockwise, Cohomological dimension, Compact-open topology, Connected space, Countable set, Curve complex, Diffeomorphism, Finitely generated group, Free group, Geometric group theory, Group action, Group theory, Homeomorphism, Hurwitz's automorphisms theorem, Identity component, Intersection number, Jakob Nielsen (mathematician), Joan Birman, Lantern relation, Locally finite space, Manifold, Mapping class group, Max Dehn, Modular group, Moduli space, Nielsen realization problem, Open problem, Orientability, Out(Fn), Presentation of a group, Representation theory, Residually finite group, Riemann surface, Singular homology, Surface (topology), Symplectic group, Teichmüller space, Thurston boundary, Tits alternative, Topological quantum field theory, Topology, Torus, Weil–Petersson metric, William Thurston, 3-manifold.
Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Annulus (mathematics)
In mathematics, an annulus (the Latin word for "little ring" is anulus/annulus, with plural anuli/annuli) is a ring-shaped object, a region bounded by two concentric circles.
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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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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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Clockwise
Two-dimensional rotation can occur in two possible directions.
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Cohomological dimension
In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations.
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Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.
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Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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Curve complex
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite type surface S, which encodes the combinatorics of simple closed curves on S.
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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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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Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
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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 theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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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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Hurwitz's automorphisms theorem
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1).
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Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.
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Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency.
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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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Joan Birman
Joan Sylvia Lyttle Birman (born May 30, 1927 in New York CityLarry Riddle. "", Biographies of Women Mathematicians, at Agnes Scott College) is an American mathematician, specializing in braid theory and knot theory.
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Lantern relation
In geometric topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface.
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Locally finite space
In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood.
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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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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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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Modular group
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
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Moduli space
In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
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Nielsen realization problem
The Nielsen realization problem is a question asked by about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by.
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Open problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (no solution for it is known).
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Orientability
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.
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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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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Residually finite group
In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every nontrivial element g in G there is a homomorphism h from G to a finite group, such that There are a number of equivalent definitions.
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Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.
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Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X).
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Surface (topology)
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
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Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.
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Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.
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Thurston boundary
In mathematics the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface.
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Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
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Topological quantum field theory
A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
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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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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Weil–Petersson metric
In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space Tg,n of genus g Riemann surfaces with n marked points.
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William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.
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3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.
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Redirects here:
Dehn-Nielsen theorem, Dehn–Nielsen theorem, Teichmuller modular group, Teichmüller modular group.
References
[1] https://en.wikipedia.org/wiki/Mapping_class_group_of_a_surface
