A collection of nonfilling multicurve complexes

We introduce a new collection of simplicial complexes associated to a connected orientable compact surface S, called k  -curve complexes, denoted by k-C(S)k-C(S), each containing vertices given by (k−1)(k−1)-simplices of the original curve complex of S  , C(S)C(S) and edges given by a restricted nonfillingness property between vertices. We prove that each complex of this collection of complexes is connected and we study their coarse geometry, in particular, we prove that the first complex, 1-C(S)1-C(S), is hyperbolic and distances in k-C(S)k-C(S) are comparable to the distances in the marking complex and therefore comparable to the length of word path in MCG(S)MCG(S). In addition, we relate the combinatorial information of 1-C(S)1-C(S) with the topology of 3-manifolds and we provide an improved bound for distance of Heegaard splittings by using the distance in the 1-curve complex, 1-C(S)1-C(S).