The moduli space of hyperbolic metrics on a compact oriented surface without boundary has a well-known symplectic structure. In this talk, I will describe a

generalization to the (infinite-dimensional) moduli space of conformally compact hyperbolic metrics on surfaces with boundary, up to diffeomorphisms fixing the

boundary. The action of boundary diffeomorphisms is Hamiltonian, making this space into an example of a Hamiltonian Virasoro space. (Joint work with Anton

Alekseev.)