Flexibility of ideal triangle groups in complex hyperbolic geometry (Q1585061)

Let \(\Gamma\) be the free product of three involutions \(\tau_i\). It can be interpreted as a group of transformations of the complex hyperbolic space \(H^2_{{\mathbb C}}\) in the following manner: Consider \(H^2_{{\mathbb C}}\) as the unit ball in \({\mathbb C}^2\). The intersections of the unit sphere \(S^3\), the boundary of \(H^2_{{\mathbb C}}\), with totally real geodesic \(2\)-planes are called \({\mathbb R}\)-circles and an inversion on an \({\mathbb R}\)-circle is, by definition, a non-trivial conformal transformation that fixes pointwise the circle. Any given \({\mathbb R}\)-circle defines a unique inversion and the involutions \(\tau_i\) are taken to be the inversions corresponding to a certain special configuration of \({\mathbb R}\)-circles. Let \(\widehat{PU(2,1)}\) be the group of orientation preserving isometries of \(H^2_{{\mathbb C}}\) (if \(H^2_{{\mathbb C}}\) is considered as an open subset of \({\mathbb C}{\mathbb P}^2\) in the standard way, this is the group generated by \(PU(2,1)\) acting by linear projective transformations and by anti-holomorphic transformations). Denote by \(\Hom_{p}(\Gamma,\widehat{PU(2,1)})\) the space of homomorphisms of \(\Gamma\) into \(\widehat{PU(2,1)}\) with \(\tau_i\) anti-holomorphic and \(\tau_i\circ\tau_j\) parabolic. The quotient space of the latter space by \(\widehat{PU(2,1)}\) acting by conjugation is called ``the ideal deformation space''. The authors define a special discrete embedding of \(\Gamma\) into the ideal deformation space, called ``the standard embedding'', using the interpretation of \(\Gamma\) described above. The main result of the paper says that there exists a neighbourhood of the standard embedding in the ideal deformation space containing only discrete embeddings. This phenomenon, opposite to the local rigidity of a discrete embedding, is considered by the authors as a local flexibility property of the triangle group \(\Gamma\).