We present a method by which torsion-free groups of automorphisms of a 2-dimensional
hyperbolic building which act simply transitively on the vertex set can be constructed, and
prove that any such group can be obtained by this construction. The method produces
groups defined by finite presentations with strong small cancellation properties, and we
prove that when the building is Fuchsian with a regular fundamental chamber, two such
groups are isomorphic if and only if there is an isomorphism taking generators to generators
and relators to relators. Using these results, we find and classify all the torsion-free
vertex-regular groups of automorphisms of Bourdon's building I55.