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Dehn Function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the ''area'' of a ''relation'' in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79–80 in ). The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive (see Theorem 2.1 in ). The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 group (mathematics), groups and topology, topological and geometry, geometric properties of spaces on which these groups Group action (mathematics), act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph (discrete mathematics), graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null-homotopic
In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariant (mathematics), invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or Spectrum (homotopy theory), spectra. Formal definition Formally, a homotopy between two continuous function (topology), continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times [0,1] \to Y from the product topology, product of the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' and ''c'' can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ''n''-dimensional systems, and most generally, to any symplectic vector space. The three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by: :\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end \begin 1 & a' & c'\\ 0 & 1 & b'\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Abelian Group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group with basis B may be described in several equivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Algorithm
In computer science, a deterministic algorithm is an algorithm that, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. Deterministic algorithms are by far the most studied and familiar kind of algorithm, as well as one of the most practical, since they can be run on real machines efficiently. Formally, a deterministic algorithm computes a mathematical function; a function has a unique value for any input in its domain, and the algorithm is a process that produces this particular value as output. Formal definition Deterministic algorithms can be defined in terms of a state machine: a ''state'' describes what a machine is doing at a particular instant in time. State machines pass in a discrete manner from one state to another. Just after we enter the input, the machine is in its ''initial state'' or ''start state''. If the machine is deterministic, this means that from this point onwards, ... [...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] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. History The notion of a Van Kampen diagram was introduced by Egbert van Kampen in 1933. This paper appeared in the same issue of American Journal of Mathematics as another paper of Van Kampen, where he proved what is now known as the Seifert–Van Kampen theorem. The main result of the paper on Van Kampen diagrams, now known as the ''van Kampen lemma'' can be deduced from the Seifert–Van Kampen theorem by applying the latter to the presentation complex of a group. However, Van Kampen did not notice it at the time and this fact was only made explicit much later (see, e.g.Aleksandr Yur'evich Ol'shanskii. ''Geometry of defining relations in groups.'' Translated from the 1989 Russia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentation Complex
In geometric group theory, a presentation complex is a 2-dimensional cell complex associated to any presentation of a group, presentation of a group (mathematics), group ''G''. The complex has a single vertex, and one loop at the vertex for each generating set of a group, generator of ''G''. There is one 2-cell for each relation in the presentation, with the boundary of the 2-cell attached along the appropriate word (group theory), word. Properties * The fundamental group of the presentation complex is the group ''G'' itself. * The universal cover of the presentation complex is a Cayley complex for ''G'', whose 1-skeleton is the Cayley graph of ''G''. * Any presentation complex for ''G'' is the 2-skeleton of an Eilenberg–MacLane space K(G,1). Examples Let G= \Z^2 be the two-dimensional integer lattice (group), lattice, with presentation : G=\langle x,y, xyx^y^\rangle. Then the presentation complex for ''G'' is a torus, obtained by gluing the opposite sides of a square, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Closure (group Theory)
In group theory, the normal closure of a subset S of a Group (mathematics), group G is the smallest normal subgroup of G containing S. Properties and description Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the intersection of all normal subgroups of G containing S: \operatorname_G(S) = \bigcap_ N. The normal closure \operatorname_G(S) is the smallest normal subgroup of G containing S, in the sense that \operatorname_G(S) is a subset of every normal subgroup of G that contains S. The subgroup \operatorname_G(S) is Generating set of a group, generated by the set S^G=\ = \ of all Conjugacy class, conjugates of elements of S in G. Therefore one can also write \operatorname_G(S) = \. Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set \varnothing is the trivial subgroup. A variety of other notations are used for the normal closure in the literature, including \langle S^G\rangle, \langle S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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