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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 1910s.〔Bruce Chandler and Wilhelm Magnus, ''The history of combinatorial group theory. A case study in the history of ideas.'' Studies in the History of Mathematics and Physical Sciences, 9. Springer-Verlag, New York, 1982. ISBN 0-387-90749-1.〕 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 Gauss–Bonnet theorem for a closed loop in the Cayley graph to conclude that such a loop must contain a large portion (more than a half) of a defining relation. A 1949 paper of Tartakovskii〔V. A. Tartakovskii, ''Solution of the word problem for groups with a k-reduced basis for k>6''. (Russian) Izvestiya Akad. Nauk SSSR. Ser. Mat., vol. 13, (1949), pp. 483–494.〕 was an immediate precursor for small cancellation theory: this paper provided a solution of the word problem for a class of groups satisfying a complicated set of combinatorial conditions, where small cancellation type assumptions played a key role. The standard version of small cancellation theory, as it is used today, was developed by Martin Greendlinger in a series of papers in early 1960s,〔Martin Greendlinger, (''Dehn's algorithm for the word problem.'' ) Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 67–83.〕〔Martin Greendlinger, (''On Dehn's algorithms for the conjugacy and word problems, with applications''. ) Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 641–677.〕〔Martin Greendlinger, (''An analogue of a theorem of Magnus.'' ) Archiv der Mathematik, vol 12 (1961), pp. 94–96.〕 who primarily dealt with the "metric" small cancellation conditions. In particular, Greendlinger proved that finitely presented groups satisfying the C'(1/6) small cancellation condition have word problem solvable by Dehn's algorithm. The theory was further refined and formalized in the subsequent work of Lyndon,〔Roger C. Lyndon, (''On Dehn's algorithm.'' ) Mathematische Annalen, vol. 166 (1966), pp. 208–228.〕 Schupp〔Paul E. Schupp, (''On Dehn's algorithm and the conjugacy problem.'' ) Mathematische Annalen, vol 178 (1968), pp. 119–130.〕 and Lyndon-Schupp,〔 who also treated the case of non-metric small cancellation conditions and developed a version of small cancellation theory for amalgamated free products and HNN-extensions. Small cancellation theory was further generalized by Alexander Ol'shanskii who developed〔Alexander Yu. Olʹshanskii, ''Geometry of defining relations in groups''. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991. ISBN 0-7923-1394-1.〕 a "graded" version of the theory where the set of defining relations comes equipped with a filtration and where a defining relator of a particular grade is allowed to have a large overlap with a defining relator of a higher grade. Olshaskii used graded small cancellation theory to construct various "monster" groups, including the Tarski monster〔A. Yu. Olshanskii, ''An infinite group with subgroups of prime orders'', Math. USSR Izv. 16 (1981), 279–289; translation of Izvestia Akad. Nauk SSSR Ser. Matem. 44 (1980), 309–321.〕 and also to give a new proof〔A. Yu. Olshanskii, ''Groups of bounded period with subgroups of prime order'', Algebra and Logic 21 (1983), 369-418; translation of Algebra i Logika 21 (1982), 553-618.〕 that free Burnside groups of large odd exponent are infinite (this result was originally proved by Adian and Novikov in 1968 using more combinatorial methods).〔P. S. Novikov, S. I. Adian, (''Infinite periodic groups. I''. ) Izvestia Akademii Nauk SSSR. Ser. Mat., vol. 32 (1968), no. 1, pp. 212–244.〕〔P. S. Novikov, S. I. Adian, ''Infinite periodic groups. II''. Izvestia Akademii Nauk SSSR. Ser. Mat., vol. 32 (1968), no. 2, pp. 251–524.〕〔P. S. Novikov, S. I. Adian. ''Infinite periodic groups. III''. Izvestia Akademii Nauk SSSR. Ser. Mat., vol. 32 (1968), no. 3, pp. 709–731.〕 Small cancellation theory supplied a basic set of examples and ideas for the theory of word-hyperbolic groups that was put forward by Gromov in a seminal 1987 monograph "Hyperbolic groups".〔M. Gromov, ''Hyperbolic Groups'', in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「small cancellation theory」の詳細全文を読む スポンサード リンク
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