数学问题-1 SL(2,C) 的表示
Oct 16

Fundamental groups and finte groups

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Let \Sigma be a closed surface of genus g . Its fundamental group \pi(\Sigma) has generators x_1,y_1,\dots, x_g,y_g satisfying the relation
(1)\ \ x_1y_1x_1^{-1}y_1^{-1}\dots x_gy_gx_g^{-1}y_g^{-1}=1.

Let G be a finite group. The question is how many homomorphisms there are form \pi(\Sigma) to G. Or equivalently how many solutions the equation (1) has in the group G. The answer is the following theorem due to A. D. Mednykh [S]:

Theorem: The number of solutions of (1) in G is
|G|^{2g-1}\sum_{\rho}(\dim \rho)^{2g-2},
where the sum runs over all the complex irreducible representation of G.

This result relates to the moduli space of flat connections evaluated in a finite group.

[S] A. N. Sengupta, A functional integral applied to topology and algebra, XIVth International Congress on mathematical physics, World Scientific, 2005, 527-532

4 Responses to “Fundamental groups and finte groups”

  1. xiphoid Says:
    十月 16th, 2007 at 6:23 下午

    Could you briefly give the idea of how to show this?

  2. sqps Says:
    十月 16th, 2007 at 10:10 下午

    Define a function f on the finite group which is 1 on the identity element and which is 0 on any other elements. The number of solutions can be write as a sumation of function f. On the other hand, f can be write as a linear combination of irreducible representations of G. By using fomula in representation theory, one obtains the result.

  3. mathtalk Says:
    十月 22nd, 2007 at 5:43 上午

    如果是其它的群, 比如一个三维流形的基本群, 有没有类似的
    结论呢?

  4. sqps Says:
    十月 22nd, 2007 at 9:42 上午

    It was solved by Frobenius for the fundamental group of a nonorientable surface. [S] What I presented above is for a orientable surface.

    I do not know whether there are some results about other groups. Maybe we can consider the fundamental group of the complement of a knot.

    Another question is to characterize the relation R such that the enumeration problem for a finite generated group with the only relation R in its generators can be solved.

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