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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ping-pong lemma ： ウィキペディア英語版 Ping-pong lemma In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group. ==History== The ping-pong argument goes back to late 19th century and is commonly attributed〔 to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently Möbius transformations of the Riemann sphere. The ping-pong lemma was a key tool used by Jacques Tits in his 1972 paper〔J. Tits. (''Free subgroups in linear groups.'' ) Journal of Algebra, vol. 20 (1972), pp. 250–270〕 containing the proof of a famous result now known as the Tits alternative. The result states that a finitely generated linear group is either virtually solvable or contains a free subgroup of rank two. The ping-pong lemma and its variations are widely used in geometric topology and geometric group theory. Modern versions of the ping-pong lemma can be found in many books such as Lyndon&Schupp,〔Roger C. Lyndon and Paul E. Schupp. ''Combinatorial Group Theory.'' Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1; Ch II, Section 12, pp. 167–169〕 de la Harpe,〔Pierre de la Harpe. (''Topics in geometric group theory.'' ) Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ISBN 0-226-31719-6; Ch. II.B "The table-Tennis Lemma (Klein's criterion) and examples of free products"; pp. 25–41.〕 Bridson&Haefliger〔Martin R. Bridson, and André Haefliger. (''Metric spaces of non-positive curvature.'' ) Grundlehren der Mathematischen Wissenschaften (Principles of Mathematical Sciences ), 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9; Ch.III.Γ, pp. 467–468〕 and others. 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「Ping-pong lemma」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.