  The string vertices of closed string field theory are subsets of the moduli
spaces of punctured Riemann surfaces that satisfy a geometric version of the
Batalin-Vilkovisky master equation. We present a homological proof of existence
of string vertices and their uniqueness up to canonical transformations. Using
hyperbolic metrics on surfaces with geodesic boundaries we give an exact
construction of string vertices as sets of surfaces with systole greater than
or equal to $L$ with $L\leq 2\, \hbox{arcsinh}\, 1$. Intrinsic hyperbolic
collars prevent the appearance of short geodesics upon sewing. The surfaces
generated by Feynman diagrams are naturally endowed with Thurston metrics:
hyperbolic on the vertices and flat on the propagators. For the classical
theory the length $L$ is arbitrary and, as $L\to \infty$ hyperbolic vertices
become the minimal-area vertices of closed string theory.
