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
hi})....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

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