Geometric Structures on Riemann Surfaces and Reidemeister Torsion

Abstract

Let Σ be a closed orientable surface of genus at least 2 and Rep(Σ,G) be the smooth part of the representation variety of homomorphisms' conjugation classes from fundamental group of Σ to Lie group G.

In this thesis, the Reidemeister torsion formulas of the representations corresponding to geometric structures in two different categories, real and complex, are clearly stated that they can be calculated through the related symplectic forms.

This thesis consists of two main parts:

In the first part, real projective structures are discussed. The deformation space B(Σ) of convex real projective structures on the surface has the Goldman coordinates in the literature and this space also contains the Teichmüller space. Using these coordinates, H.C. Kim clearly expressed the Atiyah-Bott-Goldman symplectic form ω_PSL(3,R) on the representation space Rep(PSL(3,R)). In this part, in the light of all this information, the formula that calculates the Reidemeister torsion of representations Rep(PSL(3,R)) is obtained through the symplectic form ω_PSL(3,R). In the second part, complex projective structures are considered. There is a natural holomorphic projection from CP(Σ) the space of isotopy classes of complex projective structures on the surface to the Teichmüller space. Any smooth section s of this projection yields a diffeomorhism between CP(Σ) and the cotangent bundle space T^*Teich(Σ) . There is the symplectic form ω_PSL(2,C) on CP(Σ) which is open in Rep(PSL(2,C)) and the symplectic form ω_nat on T^*Teich(Σ) . In this part, the Reidemeister torsion of the representations in CP(Σ) are expressed by ω_nat and ω_PSL(2,C) symplectic forms thanks to considered s section is Bers, Schottky, Earle and Fuchsian section, respectively. In addition, the results are applied to 3-manifolds that its boundary consisting of closed surfaces with genus at least 2.