Integral points on character varieties
Let \(M\) be a finite volume hyperbolic \(3\)-manifold with a single cusp, and let \(X(M)\) be the character variety associated to representations of its fundamental group to \(SL(2,\mathbb{C})\). It is known that the component \(X_0\) of \(X(M)\) containing the character of the faithful discrete representation is an irreducible algebraic curve, and that \(X_0\) carries much topological information about \(M\). For a quadratic imaginary number field \(\mathbb{Q}(\sqrt{-d})\), let \(O_d\) denote the ring of integers. A point on \(X_0\) whose coordinates are in \(\mathbb{Z}\) (resp. \(O_d\)) is called a \(\mathbb{Z}\)-integral point (resp. \(O_d\)-point). The authors prove that the number of \(O_d\)-points on \(X_0\) is finite for each \(d\), and that for large enough \(d\), there are no \(O_d\)-points on \(X_0\) other than possibly \(\mathbb{Z}\)-integral points. For \(\mathbb{Z}\)-points, they can say more. If a hyperbolic knot \(K\) in an integral homology \(3\)-sphere admits a \(\mathbb{Z}\)-integral point on its \(X_0\), then the point corresponds to a surjection of the knot group onto either \(\Sigma_3^*\) or \(A_4^*\), where \(\Sigma_3^*\) or \(A_4^*\) denote the central extensions by lifting the symmetric group on \(3\) letters and alternating group on \(4\) letters from \(SO(3)\) to \(SU(2)\). In fact, the existence of such surjection is controlled by Alexander polynomials. As a stronger result, in case that \(X(K)\) contains a \(\mathbb{Z}\)-integral point corresponding to an irreducible representation, the same conclusion holds, or the knot group surjects onto the trefoil knot group. Hence, the number of integral points on \(X(K)\) corresponding to an irreducible representation is finite, unless the knot group surjects onto the trefoil group. There are also examples of calculations and an application to finite Dehn surgery problems on hyperbolic knots in \(S^3\).
scientific article; zbMATH DE number 1922399