Self-avoiding random surfaces with fluctuating topology

A gas of self-avoiding surfaces with an arbitrary polynomial coupling to the gaussian curvature and an extrinsic curvature term can be realized in a three-dimensional Ising bcc lattice with only three local couplings. Similar three parameter realizations are valid also in other lattices. The relation between the crumpling transition and the roughening is discussed. It turns out that the mean area of these surfaces is proportional to its genus.