The \(\mathrm{SL}_2(\mathbb C)\)-character variety of a group \(\Gamma\) is defined as the (invariant-theoretic) quotient \(X(\Gamma) = \mathrm{Hom}(\Gamma, \mathrm{SL}_2(\mathbb C)) /\! / \Gamma\). This paper concerns the case \(\Gamma = \pi_1 (\Sigma)\), where \(\Sigma\) denotes a closed connected surface with finitely many punctures. The first result (Thm. A) states that, provided the genus \(g\) of \(\Sigma\) is at least \(2\), a semisimple representation \(\pi_1(\Sigma) \to \mathrm{SL}_2(\mathbb C)\) has finite orbit in the character variety \(X(\pi_1(\Sigma))\) with respect to the action of the mapping class group \(\mathrm{Mod}(\Sigma)\) of \(\Sigma\) if and only if it is finite, which means it has finite image. A second result (Thm. B) concerns the case of genus \(1\). In this case, a semisimple representation \(\rho\colon \pi_1(\Sigma) \to \mathrm{SL}_2(\mathbb C)\) has finite orbit in the character variety \(X(\pi_1(\Sigma))\) if and only if it is finite or special dihedral, which means it factors through the infinite dihedral group \(D_{\infty} \subset \mathrm{GL}_2(\mathbb C)\) comprising the diagonal and antidiagonal matrices in \(\mathrm{SL}_2(\mathbb C)\), and there is a non-seperating simple closed curve \(a\) such that \(\rho\) is diagonal on the complement of \(a\). A third result (Thm. C) characterizes representations \(\rho\) whose orbit under the action of \(\mathrm{Mod}(\Sigma)\) is bounded (i.e., its closure with respect to the Euclidean topology is compact) as those that are unitary up to conjugacy or, if \(g =1\), special dihedral up to conjugacy.
The main ingredients in the proof of Theorems A and B are Kronecker’s theorem on algebraic integers, clever geometric arguments, and, most importantly, an earlier theorem of the last named author and collaborators stating that a representation \(\pi_1(\Sigma) \to \mathrm{SL}_2(\mathbb C)\) whose monodromy along all simple closed loops is finite, is finite itself [A. Patel et al., Adv. Math. 386, Article ID 107800, 33 p. (2021; Zbl 1467.14087)]. Theorem C likewise relies on a result from the same paper stating that a representation \(\pi_1(\Sigma) \to\mathrm{SL}_2(\mathbb C)\) whose monodromy along all simple closed loops is elliptic or central, is unitarizable.