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Δ-hyperbolic Space
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called Gromov-hyperbolic groups. Definitions In this paragraph we give various definitions of a \delta-hyperbolic space. A metric space is said to be (Gromov-) hyperbolic if it is \delta-hyperbolic for some \delta > 0. Definition using the Gromov product Let (X,d) be a metric space. The Gromov product of two points y, z \in X with respect to a third one x \in X is defined by the formula: :(y,z)_x = \frac 1 2 \left( d(x, y) + d(x, z) - d(y, z) \right). Gromov's definition of a hyperbolic metric space is then as follows: X is \delta-hyperbolic if and only if all x,y,z,w \in X satisfy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-isometry
In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov. Definition Suppose that f is a (not necessarily continuous) function from one metric space (M_1,d_1) to a second metric space (M_2,d_2). Then f is called a ''quasi-isometry'' from (M_1,d_1) to (M_2,d_2) if there exist constants A\ge 1, B\ge 0, and C\ge 0 such that the following two properties both hold:P. de la Harpe, ''Topics in geometric group theory''. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. #For every two points x and y in M_1, the distance between their images is up to the addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End (graph Theory)
In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit–evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph. Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end. Definition and characterization Ends of graphs were defined by in terms of equivalence classes of infinite paths. A in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v_0,v_1,v_2,\dots in which each vertex appears at most once in the sequence and each two consecutive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultralimit
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and uses an ultrafilter to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion of Gromov–Hausdorff convergence of metric spaces. Ultrafilters An ultrafilter ''ω'' on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and which, given any subset ''X'' of , contains either ''X'' or . An ultrafilter ''ω'' on is ''non-principal'' if it contains no finite set. Limit of a sequence of points with respect to an ultrafilter Let ''ω'' be a non-principal ultrafilter on \mathbb N . If (x_n)_ is a sequence of points in a metric space (''X'',''d'') and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem. History See for a detailed history of combinatorial group theory. A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein Christian Felix Klein (; 25 April 1849 – 22 Ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem. History Some ideas underlying the small cancellation theory go back to the work of Max Dehn in the 1910s. Dehn proved that fundamental groups of closed orientable surfaces of genus at least two have word problem solvable by what is now called Dehn's algorithm. His proof involved drawing the Cayley graph of such a group in the hyperbolic plane and performing curvature estimates via the Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. '' Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. '' Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-isometry
In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov. Definition Suppose that f is a (not necessarily continuous) function from one metric space (M_1,d_1) to a second metric space (M_2,d_2). Then f is called a ''quasi-isometry'' from (M_1,d_1) to (M_2,d_2) if there exist constants A\ge 1, B\ge 0, and C\ge 0 such that the following two properties both hold:P. de la Harpe, ''Topics in geometric group theory''. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. #For every two points x and y in M_1, the distance between their images is up to the addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CAT Space
In mathematics, a \mathbf(k) space, where k is a real number, is a specific type of metric space. Intuitively, triangles in a \operatorname(k) space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a \operatorname(k) space, the curvature is bounded from above by k. A notable special case is k=0; complete \operatorname(0) spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard. Originally, Aleksandrov called these spaces “\mathfrak_k domain”. The terminology \operatorname(k) was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications). Definitions For a real number k, let M_k denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature k. Denote by D_k the diameter of M_k, which is \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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