Relatively extra-large Artin groups

Arye Juhász¹

¹Israel Institute of Technology, Haifa, Israel

Groups, geometry, and dynamics, Tome 12 (2018) no. 4, pp. 1343-1370

Résumé

Let n≥2 be an integer and let N be an n×n symmetric matrix with 1's on the main diagonal and natural numbers nij​=1 as off-diagonal entries. (0 is a natural number). Let X={x1​,...,xn​} and let F be the free group on X. For every non-zero off-diagonal entry nij​ of N define a word Rij​:=UV−1 in F, where U is the initial subword of (xi​xj​)nij​ of length nij​ and V is the initial subword of (xj​xi​)nij​ of length nij​, 1≤i,j≤n. Let A be the group given by the presentation 〈X∣Rij​,nij​≥2〉. A is called the Artin group defined by N, with standard generators X. Let Y={x1​,...,xk​},k<n and let NY​ be the submatrix of N corresponding to Y. Let H=〈Y〉. We call A extra-large relative to H if N subdivides into submatrices NY​,B,C and D of sizes k×k,k×l,l×k,l×l, respectively (l+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 NY​. In this work we solve the word problem for such A, show that A is torsion free and show that A has property K(π,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.

Détail

DOI : 10.4171/ggd/471

Classification : 20-XX
Mots-clés : Artin groups, Word Problem, K(π,1) property, relative presentations, Howie diagrams, small cancellation theory