The topology of the relative character varieties of a quadruply-punctured sphere. (English) Zbl 0957.57003

Authors’ abstract: “Let \(M\) be a quadruply-punctured sphere with boundary components \(A\), \(B\), \(C\), \(D\). The \(\text{SL} (2,\mathbb{C})\)-character variety of \(M\) consists of equivalence classes of homomorphisms \(\rho\) of \(\pi_1(M)\to \text{SL} (2,\mathbb{C})\) and can be identified with a quartic hypersurface in \(\mathbb{C}^7\). For fixed \(a,b,c,d\in \mathbb{C}\), the subset \(V_{a,b,c,d}\) corresponding to representations \(\rho\) with \(\text{tr} (\rho(A))= a\), \(\text{tr} (\rho(B))= b\), \(\text{tr} (\rho(C))= c\), \(\text{tr} (\rho(D))= d\) is a cubic surface in \(\mathbb{C}^3\). We determine the singular points of \(V_{a,b,c,d}\) and classify its set \(V_{a,b,c,d} (\mathbb{R})\) of \(\mathbb{R}\)-points into six topological types, at least when this set is nonsingular, \(V_{a,b,c,d} (\mathbb{R})\) 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 \(\text{SL} (2,\mathbb{R})\)”.


57M05 Fundamental group, presentations, free differential calculus
20C99 Representation theory of groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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