Projective structures on open surfaces

On monodromy of complex projective structures

“…Proof The proof of this Proposition is similar to [8]. Let c,b~Th(Qi), g, h ~ ~1 (Ki) be primitive peripheral elements of ~1 (E) (see Fig.…”

“…Some particular cases of Theorem 1 were established earlier. Under the assumption that the monodromy group p(F) is contained in SL(2, IR) Theorem 1 was proven in the preprint [8]. Suppose that p factors through a homomorphism onto a free group of rank g so that the images in SL(2,C) of free generators are loxodromic.…”

“…Namely, we connect the representation p with a Fuchsian representation r o by a special family of homomorphisms r~ : F -~ SL(2, ~), 0 < t < l, r~ = p (Theorem 2). This is done by generalizing arguments of [8] and [9]. The map from the space of all complex projective structures on S into the representation variety Hom(F, SL(2, C))/SL(2,r is open [15].…”

“…D. Hejhal also conjectured that "generic" representations into SL(2, ~) are monodromy representations. Our proof of Theorem 1 is based on ideas of [4], [7], [8] and [9].…”

“…The idea of the proof of Theorem 1 is to combine the "continuity method" of [4] with combinatorial arguments of [7], [8] using properties [9] of the representation variety Hom(F, SL(2,tlS))/SL (2,112). Namely, we connect the representation p with a Fuchsian representation r o by a special family of homomorphisms r~ : F -~ SL(2, ~), 0 < t < l, r~ = p (Theorem 2).…”