On monodromy map

Jharna D. Sengupta

International Journal of Mathematics and Mathematical Sciences, 1993, vol. 16, 1-14

Abstract:

Let Γ be a Fuchsian group acting on the upper half-plane U and having signature { p , n , 0 ; v 1 , v 2 , … , v n } ; 2 p − 2 + ∑ j = 1 n ( 1 − 1 v j ) > 0 .

Let T ( Γ ) be the Teichmüller space of Γ . Then there exists a vector bundle ℬ ( T ( Γ ) ) of rank 3 p − 3 + n over T ( Γ ) whose fibre over a point t ∈ T ( Γ ) representing Γ t is the space of bounded quratic differentials B 2 ( Γ t ) for Γ t . Let Hom ( Γ , G ) be the set of all homomorphisms from Γ into the Mbius group G .

For a given ( t , ϕ ) ∈ ℬ ( T ( Γ ) ) we get an equivalence class of projective structures and a conjugacy class of a homomorphism x ∈ Hom ( Γ , G ) . Therefore there is a well defined map Φ : ℬ ( T ( Γ ) ) → Hom ( Γ , G ) / G , Φ is called the monodromy map. We prove that the monromy map is hommorphism. The case n = 0 gives the previously known result by Earle, Hejhal Hubbard.

Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:985870

DOI: 10.1155/S0161171293000870

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