Representations of surface groups in complex hyperbolic space
Let S be a closed Riemann surface of genus \(g>1\). If \(\rho\) : \(\pi_ 1S\to PSU(1,n)\) is a representation of the surface group, the associated flat bundle over S fibered by \(B^ n\) can be trivialised by means of an equivariant map f: \(H^ 2\to B^ n\). If \(\omega\) is the natural Kähler form on \(B^ n\), let \(f^*\omega\) denote the induced form on S, and write \(c(\rho)=\int_{S}f^*\omega\). Then it is known that \(| c(\rho)| \leq 4\pi (g-1),\) and the author proves that if we have equality, then \(\rho\) leaves invariant some complex line in \(B^ n\). He deduces that the space of flat bundles such that \(| c(\rho)| -4\pi (g-1)\) is homeomorphic to the product of the Teichmüller space for S and the space of flat U(n-1)-bundles over S. The critical step in the proof is the extension of f to a map of \(\bar H^ 2\) into \(\bar B^ n\); the image of \(\partial \bar H^ 2\) in \(\bar B^ n\) is then the boundary of the required complex line.
representation of fundamental group by hyperbolic isometries
scientific article; zbMATH DE number 4107754