# Is there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cusps

If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get a surface with k punctures/cusps. Is there a concept of distance ( in Teichmuller theory/ Riemannian geometry ) which would realize this convergence ? I know that there is a concept of Teichmuller space with genus g and k geodesic boundary components and also there is a concept of Teichmuller space with genus g and k cusps with finite area metrics. But is there a concept of Teichmuller space which will unify both of them, i.e. is distance will realize the above convergence and will contain both of the above Teichmuller spaces as subspace ?

Should we use Gromov-Housdorff convergence to realize the above convergence ?

If you have a compact hyperbolic surface with geodesic boundary $\Sigma$, then you may double the surface along its boundary to get a closed hyperbolic surface $D\Sigma=\Sigma\cup_{\partial\Sigma}\Sigma$, which has an involution $\tau:D\Sigma\to D\Sigma$ which exchanges the two sides and fixes $\partial \Sigma$. One may then identify the Teichmuller space parameterizing hyperbolic structures on $\Sigma$ with geodesic boundary with the subspace of the Teichmuller space of $D\Sigma$ which is fixed under the involution $\tau$ (which as an element of the mapping class group acts by an isometric involution on Teichmuller space of $D\Sigma$).
Consider the Teichmuller space of $D\Sigma$, and the corresponding subspace associated to $\Sigma$ (the fixed point set of $\tau$). If one pinches some of the curves associated to the fixed point of $\tau$ on $D\Sigma$, then $\tau$ acts as an involution on the noded surface, and one may append the Teichmuller space of this noded surface invariant under $\tau$ to the Teichmuller space of $\Sigma$ with the Weil-Petersson completion. This gives you the moduli space you're describing of surfaces with geodesic boundary and punctures, and the Weil-Petersson metric gives a well-defined distance function on this space.
• Thanks Dr. Agol, I still have one naive question : are the "limiting" Riemann surfaces with double cusps topologically a 2- manifold ? Are they quotient of the upper half-plane $H$ ? – Analysis Now Mar 31 '11 at 14:10