  We formulate and solve a class of two-dimensional matrix gauge models
describing ensembles of non-folding surfaces covering an oriented, discretized,
two-dimensional manifold. We interpret the models as string theories
characterized by a set of coupling constants associated to worldsheet
ramification points of various orders. Our approach is closely related to, but
simpler than, the string theory describing two-dimensional Yang-Mills theory.
Using recently developed character expansion methods we exactly solve the
models for target space lattices of arbitrary internal connectivity and
topology.
