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Relevant bibliographies by topics / Combinatorial group theory. Free groups. Equations

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Academic literature on the topic 'Combinatorial group theory. Free groups. Equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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Journal articles on the topic "Combinatorial group theory. Free groups. Equations"

1

Solomon, Reed. "Ordered Groups: A Case Study in Reverse Mathematics." Bulletin of Symbolic Logic 5, no. 1 (1999): 45–58. http://dx.doi.org/10.2307/421140.

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The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson (1985) gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups (including an equivalent of ACA0), groups and semigroup conditions which imply orderability (WKL0), the orderability of free groups (RCA0), Hölder's Theorem (RCA0), Mal'tsev's classification of the order types of countable ordered groups ( – CA0)

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MANTUROV, O. V., and V. O. MANTUROV. "FREE KNOTS AND GROUPS." Journal of Knot Theory and Its Ramifications 19, no. 02 (2010): 181–86. http://dx.doi.org/10.1142/s0218216510007826.

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Free knots are a simplification of virtual knots obtained by forgetting arrow/sign information at classical crossings. First non-trivial examples of free knots were constructed recently by the second named author. By using parity considerations, we construct invariants of free knots valued in certain groups. These groups have a simple combinatorial description, the first one being the infinite dihedral group.

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3

DELGADO, MANUEL, STUART MARGOLIS, and BENJAMIN STEINBERG. "COMBINATORIAL GROUP THEORY, INVERSE MONOIDS, AUTOMATA, AND GLOBAL SEMIGROUP THEORY." International Journal of Algebra and Computation 12, no. 01n02 (2002): 179–211. http://dx.doi.org/10.1142/s0218196702000924.

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This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = <A|R> be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation <A|R> with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.

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4

Guo, Xiao Qiang, and Zheng Jun He. "The History, Classes and Presentations of Groups." Advanced Materials Research 430-432 (January 2012): 834–37. http://dx.doi.org/10.4028/www.scientific.net/amr.430-432.834.

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First we introduce the history of group theory. Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. Secondly, we give the main classes of groups: permutation groups, matrix groups, transformation groups, abstract groups and topological and algebraic groups. Finally, we give two different presentations of a group: combinatorial group theory and geometric group theory.

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5

Bibi, Mairaj, and Martin Edjvet. "Solving equations of length seven over torsion-free groups." Journal of Group Theory 21, no. 1 (2018): 147–64. http://dx.doi.org/10.1515/jgth-2017-0032.

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AbstractPrishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-free group is solvable. Here it is shown that all equations of length seven over torsion-free groups are solvable.

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Diekert, Volker, Olga Kharlampovich, and Atefeh Mohajeri Moghaddam. "SLP compression for solutions of equations with constraints in free and hyperbolic groups." International Journal of Algebra and Computation 25, no. 01n02 (2015): 81–111. http://dx.doi.org/10.1142/s0218196715400056.

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The paper is a part of an ongoing program which aims to show that the problem of satisfiability of a system of equations in a free group (hyperbolic or even toral relatively hyperbolic group) is NP-complete. For that, we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi-linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free δ-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in δ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic groups using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length N there is an SLP for another solution such that the size of the SLP is bounded by some polynomial p(s + log N) where s is the size of the system.

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7

Satoh, Takao. "On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (2021): 338–63. http://dx.doi.org/10.1017/s0013091521000171.

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AbstractIn this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$ of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_3$ has finite index in that of the Andreadakis–Johnson filtration of $\mathrm {IA}_3$.

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8

Manturov, Vassily Olegovich. "Reidemeister moves and groups." Journal of Knot Theory and Its Ramifications 24, no. 10 (2015): 1540006. http://dx.doi.org/10.1142/s0218216515400064.

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Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

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9

Shelah, Saharon. "Pcf and abelian groups." form 25, no. 5 (2013): 967–1038. http://dx.doi.org/10.1515/forum-2013-0119.

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Abstract. We deal with some pcf (possible cofinality theory) investigations mostly motivated by questions in abelian group theory. We concentrate on applications to test problems but we expect the combinatorics will have reasonably wide applications. The main test problem is the “trivial dual conjecture” which says that there is a quite free abelian group with trivial dual. The “quite free” stands for “-free” for a suitable cardinal , the first open case is . We almost always answer it positively, that is, prove the existence of -free abelian groups with trivial dual, i.e., with no non-trivial homomorphisms to the integers. Combinatorially, we prove that “almost always” there are which are quite free and have a relevant black box. The qualification “almost always” means except when we have strong restrictions on cardinal arithmetic, in fact restrictions which hold “everywhere”. The nicest combinatorial result is probably the so-called “Black Box Trichotomy Theorem” proved in ZFC. Also we may replace abelian groups by R-modules. Part of our motivation (in dealing with modules) is that in some sense the improvement over earlier results becomes clearer in this context.

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10

FARAMARZI SALLES, ASADOLLAH, and HASSAN KHOSRAVI. "A COMBINATORIAL PROPERTY OF BURNSIDE VARIETY OF GROUPS OF FINITE EXPONENT." Journal of Algebra and Its Applications 08, no. 06 (2009): 845–53. http://dx.doi.org/10.1142/s0219498809003680.

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Let w be a word in the free group on the set {x1, …, xn} and let [Formula: see text] be the variety of groups defined by the law w = 1. Define [Formula: see text], to be the class of all groups G in which for all infinite subsets Y1, …, Yn, there exist yi ∈ Yi such that w(y1, …, yn) = 1 and define [Formula: see text] (respectively [Formula: see text]) to be the class of all groups G in which for all infinite subset Y (respectively for all m-element subset Y) there exist n distinct elements y1, …, yn ∈ Y, such that w(y1, …, yn) = 1. In this paper we prove that [Formula: see text] and [Formula: see text] for some positive integer m, where w is a certain word and [Formula: see text] is the class of finite groups.

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Books on the topic "Combinatorial group theory. Free groups. Equations"

1

Flannery, D. L. (Dane Laurence), 1965-, ed. Algebraic design theory. American Mathematical Society, 2011.

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2

Jacques, Sauloy, and Singer Michael F. 1950-, eds. Galois theories of linear difference equations: An introduction. American Mathematical Society, 2016.

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3

I, Arnolʹd V. Experimental mathematics. MSRI Mathematical Sciences Research Institute, 2015.

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Shpilrain, Vladimir, Alexander Mikhalev, and Jie-tai Yu. Combinatorial Methods: Free Groups, Polynomials, and Free Algebras. Springer, 2014.

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Combinatorial Methods: Free Groups, Polynomials, and Free Algebras (CMS Books in Mathematics). Springer, 2003.

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Exponential Genus Problems in One-relator Products of Groups (Memoirs of the American Mathematical Society). American Mathematical Society, 2007.

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Book chapters on the topic "Combinatorial group theory. Free groups. Equations"

1

Kharlampovich, Olga, and Alexei G. Myasnikov. "Equations and Fully Residually Free Groups." In Combinatorial and Geometric Group Theory. Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-7643-9911-5_8.

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Morar, Pavel V., and Artem N. Shevlyakov. "Algebraic Geometry over the Additive Monoid of Natural Numbers: Systems of Coefficient Free Equations." In Combinatorial and Geometric Group Theory. Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-7643-9911-5_11.

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3

Lyndon, Roger C., and Paul E. Schupp. "Free Groups and Their Subgroups." In Combinatorial Group Theory. Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-61896-3_1.

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Weidmann, Richard. "Generating Tuples of Virtually Free Groups." In Combinatorial and Geometric Group Theory. Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-7643-9911-5_13.

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Frenkel, Elizaveta, Alexei G. Myasnikov, and Vladimir N. Remeslennikov. "Regular Sets and Counting in Free Groups." In Combinatorial and Geometric Group Theory. Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-7643-9911-5_4.

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Hock, Martin, and Boaz Tsaban. "Solving Random Equations in Garside Groups Using Length Functions." In Combinatorial and Geometric Group Theory. Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-7643-9911-5_6.

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7

Collins, Donald J. "PEAK REDUCTION AND AUTOMORPHISMS OF FREE GROUPS AND FREE PRODUCTS." In Combinatorial Group Theory and Topology. (AM-111), edited by S. M. Gersten and John R. Stallings. Princeton University Press, 1987. http://dx.doi.org/10.1515/9781400882083-006.

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Stallings, John R. "GRAPHICAL THEORY OF AUTOMORPHISMS OF FREE GROUPS." In Combinatorial Group Theory and Topology. (AM-111), edited by S. M. Gersten and John R. Stallings. Princeton University Press, 1987. http://dx.doi.org/10.1515/9781400882083-005.

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Gersten, S. M. "NONSINGULAR EQUATIONS OF SMALL WEIGHT OVER GROUPS." In Combinatorial Group Theory and Topology. (AM-111), edited by S. M. Gersten and John R. Stallings. Princeton University Press, 1987. http://dx.doi.org/10.1515/9781400882083-007.

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10

Baumslag, Gilbert. "Free groups, the calculus of presentations and the method of Reidemeister and Schreier." In Topics in Combinatorial Group Theory. Birkhäuser Basel, 1993. http://dx.doi.org/10.1007/978-3-0348-8587-4_3.

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Conference papers on the topic "Combinatorial group theory. Free groups. Equations"

1

Lynett, Patrick, Philip L. F. Liu, Hwung-Hweng Hwung, and Wen-Son Ching. "Multi-Layer Modeling of Wave Groups From Deep to Shallow Water." In ASME 2003 22nd International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2003. http://dx.doi.org/10.1115/omae2003-37087.

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A set of model equations for water wave propagation is derived by piecewise integration of the primitive equations of motion through N arbitrary layers. Within each layer, an independent velocity profile is determined. With N separate velocity profiles, matched at the interfaces of the layers, the resulting set of equations have N+1 free parameters, allowing for an optimization with known analytical properties of water waves. The optimized two-layer model equations show good linear wave characteristics up to kh ≈8, while the second-order nonlinear behavior is well captured to kh ≈6. The three-layer model shows good linear accuracy to kh ≈14, and the four layer to kh ≈20. A numerical algorithm for solving the model equations is developed and tested against nonlinear deep-water wave-group experiments, where the kh of the carrier wave in deep water is around 6. The experiments are set up such that the wave groups, initially in deep water, propagate up a constant slope until reaching shallow water. The overall comparison between the multi-layer model and the experiment is quite good, indicating that the multi-layer theory has good nonlinear, as well has linear, accuracy for deep-water waves.

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