A non-trivial family of trivial bundles with complex hyperbolic structure

Abstract

In $\mathrm{PU}(2,1)$, the group of holomorphic isometries of the complex hyperbolic plane, we study the space of involutions $R_1, R_2, R_3, R_4, R_5$ satisfying $R_5R_4R_3R_2R_1=1$, where $R_1$ is a reflection in a complex geodesic and the other $R_i$'s are reflections in points of the complex hyperbolic plane. We show that this space modulo $\mathrm{PU}(2,1)$-conjugation is bending-connected and has dimension $4$. Using this, we construct a $4$-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle with vanishing Euler number over the sphere with $5$ cone points of angle $\pi$. Bending-connectedness here means that we can naturally deform the geometric structure, like Dehn twists in Teichm\"uller theory. Additionally, finding complex hyperbolic disc orbibundles with vanishing Euler number is a hard problem, originally conjectured by W. Goldman and Y. Eliashberg and solved by S. Anan'in and N. Gusevskii, and we produce a simpler and more straightforward construction for them.