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Asymptotic Dimension
In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monograph ''Asymptotic invariants of infinite groups'' in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture. Asymptotic dimension has important applications in geometric analysis and index theory. Formal definition Let X be a metric space and n\ge 0 be an integer. We say that \operatorname(X)\le n if for every R\ge 1 there exists a uniformly bounded cover \mathcal U of X such that every closed R-ball in X intersects at most n+1 subsets from \mathcal U. Here 'uniformly bounded' means that \sup_ \operatorname(U) <\infty . We then define the ''asymptotic dimension'' |
Metric Geometry
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] |
Real Tree
In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the simplest examples of Gromov hyperbolic spaces. Definition and examples Formal definition A metric space X is a real tree if it is a geodesic space where every triangle is a tripod. That is, for every three points x, y, \rho \in X there exists a point c = x \wedge y such that the geodesic segments rho,x rho,y/math> intersect in the segment rho,c/math> and also c \in ,y/math>. This definition is equivalent to X being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). Real trees can also be characterised by a topological property. A metric space X is a real tree if for any pair of points x, y \in X all topological embeddings \sigma of the segment ,1/math> into X such that \sigma(0) = x, \, \sigma(1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Out(Fn)
In mathematics, Out(''Fn'') is the outer automorphism group of a free group on ''n'' generators. These groups play an important role in geometric group theory. Outer space Out(''Fn'') acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles. Definition A point of the outer space is essentially an \R-graph ''X'' homotopy equivalent to a bouquet of ''n'' circles together with a certain choice of a free homotopy class of a homotopy equivalence from ''X'' to the bouquet of ''n'' circles. An \R-graph is just a weighted graph with weights in \R. The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3. A more descriptive view avoiding the homotopy equivalence ''f'' is the following. We may fix an identification of the fundamental gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Class Group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Motivation Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite intersect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabilizer subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre. Definition A graph of groups over a graph is an assignment to each vertex of of a group and to each edge of of a group as well as monomorphisms and mapping into the groups assigned to the vertices at its ends. Fundamental group Let be a spanning tree for and define the fundamental group to be the group generated by the vertex groups and elements for each edge of with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, ''F'' is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group. The Thompson groups, and ''F'' in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups ''T'' and ''V'' are (rare) examples of infinite but finitely-presented simple groups. The group ''F'' is not simple but its derived subgroup 'F'',''F''is and the quotient of ''F'' by its derived subgroup is the free abelian group of rank 2. ''F'' is totally ordered, has exponential growth, and does not contain a subg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Hyperbolic
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume. Intuitive definition A group ''G'' is relatively hyperbolic with respect to a subgroup ''H'' if, after contracting the Cayley graph of ''G'' along ''H''- cosets, the resulting graph equipped with the usual graph metric becomes a δ-hyperbolic space and, moreover, it satisfies a technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place. Formal definition Given a finitely generated group ''G'' with Cayley graph ''Γ''(''G'') equipped with the path metric and a subgroup ''H'' of ''G'', one can construct the coned off Cayley graph \hat(G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word-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. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others. Definition Let G be a finitely g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric (mathematics)
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] |
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] |
