Invariants of translation surfaces

We definite invariants of translation surfaces which refine Veech groups. T hese aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Vecch group cannot even share a common tree of bala nced affine coverings. We also show that there exist translation surfaces o f isomorphic Veech groups which cannot affinely cover any common surface. W e also extend a result of Gutkin and Judge and thereby give the first examp les of noncompact Fuchsian groups which cannot appear as Veech groups. We g ive an infinite family of these.