簡易檢索 / 詳目顯示

回結果列表
研究生： 蘇 爽
Su, Shuang
論文名稱： 一個從含有單一關係元之有限生成群到SL(2,C)的同態空間之光滑性質的充分條件
A sufficient condition of smoothness of homomorphism space of a finitely generated group with one relator to SL(2,C)
指導教授： 何南國
Ho, Nan-Kuo
口試委員: 吳思曄
Wu, Siye
蕭欽玉
Hsiao, Chin-Yu
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 36
中文關鍵詞： 表現簇 、特徵簇 、模空間
外文關鍵詞： Representation variety, Character variety, Moduli space
相關次數： 點閱：3  下載：0 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 在這篇論文裡，我們描述了從有限循環群到SL(2,C)的同態空間並證明了
    在這個同態空間中，幾乎所有點都是光滑的。接著，我們考慮了一個更一般
    的情形，從含有單一關係元之有限生成群到SL(2,C)的同態空間。我們找出一
    個在此同態空間上的光滑性的充分條件。更進一步的，受到這兩個例子計算
    過程的啟發，我們發現上述之結論可以推廣到從含有單一關係元之有限生成
    群到SL(n,C)之同態空間的情況。

    In this thesis, we describe the homomorphism space of a finite cyclic group
    to SL(2,C) and show that almost all it's points are smooth points. Then, we
    consider a more general case, the homomorphism space of a finitely presented
    group with a single relator to SL(2,C). We give a sufficient condition of the
    smoothness of the homomorphism space. Moreover, inspired by the calculation
    process of these two cases, we find out that the conclusions above can be
    generalized to the case of homomorphism space of a finitely presented group
    with a single relator to SL(n,C) for arbitrary n.

    1 Introduction 5 2 The homomorphism space 6 3 The Fox derivation 9 4 Lyndon's sequence 12 4.1 Lyndon's Simple identity theorem....12 4.2 Lyndon's sequence of a finitely presented group with a single relator....14 5 The homomorphism space from Z_p to SL(2;C) 16 5.1 Description of Hom(\varPi;G)....17 5.2 Calculation of dim(ker \tensor*[^3]{\check{M}}{^2})....10 5.3 Calculation of dim H^2(\varPi, g_{Ad \circ \phi})....22 5.4 The main theorem....23 6 Homomorphism space of group with a single relator to SL(2,C) 17 7 Generalization 29 8 Appendix 33

    [1] S. Eilenberg and S. MacLane, Cohomology theory in abstract groups, I, Ann.
    Math. 48, 51-78, 1947.

    [2] C. Florentino and S. Lawton, Singularities of free group character varieties, Pacific J.
    Math. 260, 149-179, 2012.

    [3] W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv.
    Math. 54, 200-225, 1984.

    [4] W.M. Goldman, Topological components of spaces of representations, Invent.
    Math. 93, 557-607, 1988.

    [5] W.M. Goldman and J.J. Millson, Deformation of flat bundles over K ahler manifolds,
    in: Geometry and topology (Athens, GA, 1985), eds. C. McCrory and T.Shifrin,
    129-145, Lecture Notes in Pure and Appl. Math. 105, Dekker, New York,
    1987.

    [6] N.-K. Ho and C.-C.M. Liu, Connected Components of the Space of Surface Group
    Representations, IMRN 44, 2359-2372, 2003.

    [7] N.-K. Ho and C.-C.M. Liu, Connected Components of the Space of Surface Group
    Representations II, IMRN 16, 959-979, 2005.

    [8] G. Harris and C. Martin, The roots of a polynomial vary continuously as a function of the coefficients,
    Proc. Amer. Math. Soc. 100, 390-392, 1987.

    [9] N.-K. Ho, G. Wilkin and S. Wu, Condition of smoothness of moduli spaces of flat connections and of representation varieties,
    Math. Z. 293, 1-23, 2019.

    [10] D.L. Johnson, Presentations of groups, London Math. Soc. Student Texts, 15,
    Cambridge Univ. Press, Cambridge, 1990.

    [11] S. Lawton, Algebraic independence in SL(3,C) character varieties of free group,
    J. Algebra 324, 1383-1391, 2010.

    [12] S. Lawton and A.S. Sikora, Varieties of characters, Algebras and Representation Theory,
    133-1141, 2017.

    [13] J. Li, The space of surface group representations, Manuscripta Math. 78, 223-243, 1993.

    [14] R.C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. Math. 52, 650-665, 1950.

    [15] R.C. Lyndon and P.E. Schupp, Combinatorial group theory, Berlin-Heidelberg-New York, Springer, 2001.

    [16] A.S. Sikora, Character varieties, Trans. Amer. Math. Soc. 364, 5173-5208, 2012

    QR CODE