The Topology of the Relative Character Varieties of a Quadruply-Punctured Sphere

Abstract

Let M be a quadruply-punctured sphere with boundary components A, B,C, D. The SL(2, )-character variety of M consists of equivalence classes of homomorphisms p of ρ of π₁ (M) SL(2, ) and can be identified with a quartic hypersurface in ( ⁷. For fixed a, b, c, d ∊ (, the subset V_a,b,c,d corresponding to representations ρ with tr(ρ(A)) = a, tr(ρ(B)) = b, tr(ρ(C)) = c, tr(ρ(D)) = d is a cubic surface in (. We determine the singular points of V_a,b,c,d and classify its set V_a,b,c,d(R) of R-points into six topological types, at least when this set is nonsingular. V_a,b,c,d(R) contains a compact component if and only if −2 < a, b, c, d < 2. For certain values of (a, b, c, d), this component corresponds to representations in SL(2, R).

Keywords:

• surface
• fundamental group
• charact er variety
• represe ntation variety