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Baumslag–Gersten Group
In the mathematical subject of geometric group theory, the Baumslag–Gersten group, also known as the Baumslag group, is a particular one-relator group exhibiting some remarkable properties regarding its finite quotient groups, its Dehn function and the complexity of its word problem. The group is given by the presentation : G=\langle a,t \mid a^=a^2\rangle =\langle a, t \mid (t^a^t) a (t^ at)=a^2 \rangle Here exponential notation for group elements denotes conjugation, that is, g^h=h^gh for g, h\in G. History The Baumslag–Gersten group ''G'' was originally introduced in a 1969 paper of Gilbert Baumslag, as an example of a non-residually finite one-relator group with an additional remarkable property that all finite quotient groups of this group are cyclic. Later, in 1992, Stephen Gersten showed that ''G'', despite being a one-relator group given by a rather simple presentation, has the Dehn function growing very quickly, namely faster than any fixed iterate of the ... [...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] |
Residually Finite Group
{{unsourced, date=September 2022 In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a finite group, such that :h(g) \neq 1.\, There are a number of equivalent definitions: *A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element. *A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial. *A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial. *A group is residually finite if and only if it can be embedded inside the direct product of a family of finite groups. Examples Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic-by-finite groups, finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup Distortion
In geometric group theory, a discipline of mathematics, subgroup distortion measures the extent to which an overgroup can reduce the complexity of a group's word problem. Like much of geometric group theory, the concept is due to Misha Gromov, who introduced it in 1993. Formally, let generate group , and let be an overgroup for generated by . Then each generating set defines a word metric on the corresponding group; the distortion of in is the asymptotic equivalence class of the function R\mapsto\frac\text where is the ball of radius about center in and is the diameter of . A subgroup with bounded distortion is called undistorted, and is the same thing as a quasi-isometrically embedded subgroup. Examples For example, consider the infinite cyclic group , embedded as a normal subgroup of the Baumslag–Solitar group . With respect to the chosen generating sets, the element b^=a^nba^ is distance from the origin in , but distance from the origin in . In part ... [...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] |
Mahan Mitra
Mahan Mj (born Mahan Mitra (Bengali: মহান মিত্র), 5 April 1968), also known as Mahan Maharaj and Swami Vidyanathananda, is an Indian mathematician and monk of the Ramakrishna Order. He is currently Professor of Mathematics at the Tata Institute of Fundamental Research in Mumbai. He is a recipient of the 2011 Shanti Swarup Bhatnagar Award in Mathematical Sciences and the Infosys Prize 2015 for Mathematical Sciences. He is best known for his work in hyperbolic geometry, geometric group theory, low-dimensional topology and complex geometry. Early education Mahan Mitra studied at St. Xavier's Collegiate School, Calcutta, till Class XII. He then entered the Indian Institute of Technology Kanpur, with an All India Rank (AIR) of 67 in the Joint Entrance Examination, where he initially chose to study electrical engineering but later switched to mathematics. He graduated with a Masters in mathematics from IIT Kanpur in 1992. Career Mahan Mitra joined the PhD pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic-case Complexity
Generic-case complexity is a subfield of computational complexity theory that studies the complexity of computational problems on "most inputs". Generic-case complexity is a way of measuring the complexity of a computational problem by neglecting a small set of unrepresentative inputs and considering worst-case complexity on the rest. Small is defined in terms of asymptotic density. The apparent efficacy of generic case complexity is because for a wide variety of concrete computational problems, the most difficult instances seem to be rare. Typical instances are relatively easy. This approach to complexity originated in combinatorial group theory, which has a computational tradition going back to the beginning of the last century. The notion of generic complexity was introduced in a 2003 paper,I. Kapovich, A. Myasnikov, P. Schupp and V. Shpilrain, Generic case complexity, decision problems in group theory and random walks', J. Algebra, vol 264 (2003), 665–694. where authors show ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Function Arithmetic
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic,C. Smoryński, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Friedman's Research on the Foundations of Mathematics'' (1985), Studies in Logic and the Foundations of Mathematics vol. 117. is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, ''x''''y'', together with induction for formulas with bounded quantifiers. EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic. Definition EFA is a system in first order logic (with equality). Its language contains: *two constants 0, 1, *three binary operations +, ×, exp, with exp(''x'',''y'') usually written as ''x''''y'', *a binary relation symbol < (This is not really necessa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugacy Problem
In abstract algebra, the conjugacy problem for a group ''G'' with a given presentation is the decision problem of determining, given two words ''x'' and ''y'' in ''G'', whether or not they represent conjugate elements of ''G''. That is, the problem is to determine whether there exists an element ''z'' of ''G'' such that :y = zxz^.\,\! The conjugacy problem is also known as the transformation problem. The conjugacy problem was identified by Max Dehn in 1911 as one of the fundamental decision problems in group theory; the other two being the word problem and the isomorphism problem. The conjugacy problem contains the word problem as a special case: if ''x'' and ''y'' are words, deciding if they are the same word is equivalent to deciding if xy^ is the identity, which is the same as deciding if it's conjugate to the identity. In 1912 Dehn gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The outer automorphism group is usually denoted . If is trivial and has a trivial center, then is said to be complete. An automorphism of a group which is not inner is called an outer automorphism. The cosets of with respect to outer automorphisms are then the elements of ; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, , the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering as a subgroup of the symmetric group, , conjugation by any odd permutation is an oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal are Louigi Addario-Berry and Eyal Goren. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antoni ... References External links * University of Toronto Press academic journals Mathematics journals Publications established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Canada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-Hopfian Group
In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf. Formal definition A group ''G'' is called co-Hopfian if whenever \varphi:G\to G is an injective group homomorphism then \varphi is surjective, that is \varphi(G)=G.P. de la Harpe''Topics in geometric group theory''.Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ; p. 58 Examples and non-examples *Every finite group ''G'' is co-Hopfian. *The infinite cyclic group \mathbb Z is not co-Hopfian since f:\mathbb Z\to \mathbb Z, f(n)=2n is an injective but non-surjective homomorphism. *The additive group of real numbers \mathbb R is not co-Hopfian, since \mathbb R is an infinite-dimensional vector space over \mathbb Q and therefore, as a group \mathbb R\cong \mathbb R\times \mathbb R. *The additive group of rational numbers \mathbb Q and the quotient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopfian Group
In mathematics, a Hopfian group is a group ''G'' for which every epimorphism :''G'' → ''G'' is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group ''G'' is co-Hopfian if every monomorphism :''G'' → ''G'' is an isomorphism. Equivalently, ''G'' is not isomorphic to any of its proper subgroups. Examples of Hopfian groups * Every finite group, by an elementary counting argument. * More generally, every polycyclic-by-finite group. * Any finitely generated free group. * The group Q of rationals. * Any finitely generated residually finite group. * Any word-hyperbolic group. Examples of non-Hopfian groups * Quasicyclic groups. * The group R of real numbers. * The Baumslag–Solitar group ''B''(2,3). Properties It was shown by that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian. Unlike the undecidability of many properties of groups th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
