幾何群論

幾何群論是數學中的一個領域，是群論的一個分支。幾何群論通過探索幾何群的代數性質，還有這些群的群作用的空間中拓撲和幾何性質之間的聯繫來研究有限生成群（這裏的幾何群指可以用幾何上的對稱性或某些空間的連續變換來生成的群）。

幾何群論中的另一個重要思想是將有限生成群本身視為幾何對象。這通常是通過研究群的凱萊圖來完成的，除了圖論的性質以外之外，群的凱萊圖通過詞度量方法，也具有度量空間的結構，。

幾何群論是一個比較新的領域，在 20 世紀 80 年代末和 90 年代初才成為一個獨立的數學分支。幾何群論與低維拓撲、雙曲幾何、代數拓撲、計算群論和微分幾何密切相關。同時，它也與複雜性理論、數理邏輯、李群及其離散子群的研究、動力系統、概率論、 K 理論和其他數學領域也有實際聯繫。

皮埃爾·德·拉·哈普 ( Pierre de la Harpe ) 在其著作《幾何群論》 (Topics in Geometric Group Theory ) 的序言中寫道：「我相信，人類用對於對稱性和群的迷戀來應對生活挫折：我們喜歡認出（圖形中的）對稱性，這讓我們覺得：我們能夠識別出比我們看到的更多的東西。從這個意義上說，幾何群論的研究是文化的一部分，我想起喬治·德·拉姆（瑞士數學家）在許多場合做過的幾件事，比如教授數學、背誦馬拉美（Stéphane Mallarmé, 法國詩人），或者和朋友交流」。 ^[1] ^:3

幾何群論源於組合群論，它主要通過分析群表示來研究離散群的性質，群表示將群表達為自由群的商； 1880 年代初期，費利克斯·克萊因 (Felix Klein) 的學生瓦爾特·馮·戴克 ( Walther von Dyck ) 首次系統地研究了這一領域， ^[2]威廉·羅恩·漢密爾頓 (William Rowan Hamilton) 在更早通過研究二十面體微積分學性質發現了幾何群論。他研究了二十面體對稱群十二面體圖。目前，組合群論作為一個領域在很大程度上被幾何群論所包含。此外，「幾何群論」通常包括使用概率、測度論、算術、分析和傳統組合群論外的其他方法來研究離散群。

幾何群論作為數學的一個獨特領域的出現通常可以追溯到1980年代末和1990年代初。它受到米哈伊爾·格羅莫夫1987 年的專題論文《雙曲群》^[3]的啟發，該專題論文引入了雙曲群（也稱為字雙曲群word hyperbolic group或格羅莫夫雙曲群或負曲群）的概念，它抓住了有限生成群具有大規模負曲率的概念，以及他隨後的專題論文《無限群的漸近不變量》(Asymptotic Invariants of Infinite Groups)^[4]，概述了格羅莫夫所理解離散群直到擬等距的程序。格羅莫夫的工作對離散群的研究產生了變革性的影響^[5]^[6]^[7]，不久之後「幾何群論」一詞很快就出現了。

1990 年代和 2000 年代幾何群論的方向和發展包括：

格羅莫夫（Gromov）研究群的擬等距性質的計劃。
字雙曲群和相對雙曲群(相對雙曲群（英語：relatively hyperbolic group）)的理論。這裏一個特別重要的發展是 Zlil Sela 在1990年代的工作，解決了字雙曲群的同構問題^[8]。相對雙曲群的概念最初由格羅莫夫在1987年^[3] 引入，並在1990年代由 Farb^[9]和 Brian Bowditch^[10]完善。相對雙曲群的研究在2000年代得到了重視。
與數理邏輯的相互作用和自由群一階理論的研究。由於 Sela^[11] 以及 Olga Kharlampovich 和 Alexei Myasnikov 的工作，著名的塔斯基猜想 (Tarski猜想)取得了特別重要的進展^[12]。極限群(limit groups)的研究和非交換代數幾何的語言和機制的介紹獲得了突出地位。

在幾何群論中經常研究以下例子：

乒乓引理，一種將群展示為自由積的有用方法
順應組

P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6, ISBN 0-226-31721-8.
Stillwell, John (2002), Mathematics and its history, Springer, p. 374, ISBN 978-0-387-95336-6
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, vo. 9. Springer-Verlag, New York, 1982.
Greendlinger, Martin (1960). "Dehn's algorithm for the word problem". Communications on Pure and Applied Mathematics. 13 (1): 67–83. doi:10.1002/cpa.3160130108.
Greendlinger, Martin (1961). "An analogue of a theorem of Magnus". Archiv der Mathematik. 12 (1): 94–96. doi:10.1007/BF01650530. S2CID 120083990.
Roger Lyndon and Paul Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000.
J.-P. Serre, Trees. Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN 3-540-10103-9.
Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.
Mikhail Gromov, "Asymptotic invariants of infinite groups", in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.
Iliya Kapovich and Nadia Benakli. Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002.
Brian Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex. European Congress of Mathematics, pp. 103–115, Eur. Math. Soc., Zürich, 2005.

[1]
P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6, ISBN 0-226-31721-8.
[2]
Stillwell, John, Mathematics and its history, Springer: 374, 2002, ISBN 978-0-387-95336-6
[3]
Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.
[4]
Mikhail Gromov, "Asymptotic invariants of infinite groups", in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.
[5]
Iliya Kapovich and Nadia Benakli. Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002. From the Introduction:" In the last fifteen years geometric group theory has enjoyed fast growth and rapidly increasing influence. Much of this progress has been spurred by remarkable work of M. L. Gromov [in Essays in group theory, 75–263, Springer, New York, 1987; in Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, Cambridge Univ. Press, Cambridge, 1993], who has advanced the theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups)."
[6]
Brian Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex. European Congress of Mathematics, pp. 103–115, Eur. Math. Soc., Zürich, 2005. From the Introduction:" Much of this can be viewed in the context of geometric group theory. This subject has seen very rapid growth over the last twenty years or so, though of course, its antecedents can be traced back much earlier. [...] The work of Gromov has been a major driving force in this. Particularly relevant here is his seminal paper on hyperbolic groups [Gr]."
[7]
Elek, Gabor. The mathematics of Misha Gromov. Acta Mathematica Hungarica. 2006, 113 (3): 171–185. S2CID 120667382. doi:10.1007/s10474-006-0098-5 . p. 181 "Gromov's pioneering work on the geometry of discrete metric spaces and his quasi-isometry program became the locomotive of geometric group theory from the early eighties."
[8]
Sela, Zlil. The isomorphism problem for hyperbolic groups. I. Annals of Mathematics. (2). 1995, 141 (2): 217–283. JSTOR 2118520. MR 1324134. doi:10.2307/2118520.
[9]
Farb, Benson. Relatively hyperbolic groups. Geometric and Functional Analysis. 1998, 8 (5): 810–840. MR 1650094. S2CID 123370926. doi:10.1007/s000390050075.
[10]
Bowditch, Brian H. Treelike Structures Arising from Continua and Convergence Groups. Memoirs American Mathematical Society 662. American Mathematical Society. 1999. ISBN 978-0-8218-1003-3.
[11]
Zlil Sela, Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87–92, Higher Ed. Press, Beijing, 2002.
[12]
Kharlampovich, Olga; Myasnikov, Alexei. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements of the American Mathematical Society. 1998, 4 (14): 101–8. MR 1662319. doi:10.1090/S1079-6762-98-00047-X .

[1] [1]
P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6, ISBN 0-226-31721-8.

[stillwell374-2] [2]
Stillwell, John, Mathematics and its history, Springer: 374, 2002, ISBN 978-0-387-95336-6

[M._Gromov,_1987,_pp._75–263-3] [3]
Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.

[4] [4]
Mikhail Gromov, "Asymptotic invariants of infinite groups", in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.

[5] [5]
Iliya Kapovich and Nadia Benakli. Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002. From the Introduction:" In the last fifteen years geometric group theory has enjoyed fast growth and rapidly increasing influence. Much of this progress has been spurred by remarkable work of M. L. Gromov [in Essays in group theory, 75–263, Springer, New York, 1987; in Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, Cambridge Univ. Press, Cambridge, 1993], who has advanced the theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups)."

[6] [6]
Brian Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex. European Congress of Mathematics, pp. 103–115, Eur. Math. Soc., Zürich, 2005. From the Introduction:" Much of this can be viewed in the context of geometric group theory. This subject has seen very rapid growth over the last twenty years or so, though of course, its antecedents can be traced back much earlier. [...] The work of Gromov has been a major driving force in this. Particularly relevant here is his seminal paper on hyperbolic groups [Gr]."

[7] [7]
Elek, Gabor. The mathematics of Misha Gromov. Acta Mathematica Hungarica. 2006, 113 (3): 171–185. S2CID 120667382. doi:10.1007/s10474-006-0098-5 . p. 181 "Gromov's pioneering work on the geometry of discrete metric spaces and his quasi-isometry program became the locomotive of geometric group theory from the early eighties."

[8] [8]
Sela, Zlil. The isomorphism problem for hyperbolic groups. I. Annals of Mathematics. (2). 1995, 141 (2): 217–283. JSTOR 2118520. MR 1324134. doi:10.2307/2118520.

[9] [9]
Farb, Benson. Relatively hyperbolic groups. Geometric and Functional Analysis. 1998, 8 (5): 810–840. MR 1650094. S2CID 123370926. doi:10.1007/s000390050075.

[10] [10]
Bowditch, Brian H. Treelike Structures Arising from Continua and Convergence Groups. Memoirs American Mathematical Society 662. American Mathematical Society. 1999. ISBN 978-0-8218-1003-3.

[11] [11]
Zlil Sela, Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87–92, Higher Ed. Press, Beijing, 2002.

[12] [12]
Kharlampovich, Olga; Myasnikov, Alexei. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements of the American Mathematical Society. 1998, 4 (14): 101–8. MR 1662319. doi:10.1090/S1079-6762-98-00047-X .

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幾何群論

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