## 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 $$\text{SL} (2,\mathbb{R})$$”.

### MSC:

 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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### References:

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