Relatively extra-large Artin groups

Let n >= 2 be an integer and let N be an n x n symmetric matrix with l's on the main diagonal and natural numbers n(ij) not equal 1 as off-diagonal entries. (0 is a natural number). Let X = {x(1), ..., xn} and let F be the free group on X. For every non-zero off-diagonal entry n(ij) of N define a word R-ij := UV-1 in F, where U is the initial subword of (x(i) x(j))(nij )of length n(ij) and V is the initial subword of (x(j) x(i))(n) ij of length nu ,1 <= j <= n. Let A be the group given by the presentation < X vertical bar R-ij, nu(ij) >= 2 >. A is called the Artin group defined by N, with standard generators X. Let Y = {x(1), ..., x(k)}, k < n and let N-Y be the submatrix of N corresponding to Y. Let H = < Y >. We call A extra-large relative to H if N subdivides into submatrices N-Y, B, C and D of sizes k x k, k x l, l x k, l x l, respectively (1 + k = n) such that every non zero element of B and C is at least 4 and every off-diagonal non-zero entry of D is at least 3. No condition on N-Y. In this work we solve the word problem for such A, show that A is torsion free and show that A has property K(pi, 1), provided that H has these properties, correspondingly. We also compute the homology and cohomology of A, relying on that of H. The two main tools used are Howie diagrams corresponding to relative presentations of A with respect to H and small cancellation theory with mixed small cancellation conditions.