Relatively extra-large Artin groups

  • Arye Juhász

    Israel Institute of Technology, Haifa, Israel


Let be an integer and let be an symmetric matrix with 1's on the main diagonal and natural numbers as off-diagonal entries. (0 is a natural number). Let and let be the free group on . For every non-zero off-diagonal entry of define a word in , where is the initial subword of of length and is the initial subword of of length , . Let be the group given by the presentation . is called the Artin group defined by , with standard generators . Let and let be the submatrix of corresponding to . Let . We call extra-large relative to if subdivides into submatrices and of sizes , respectively such that every non zero element of and is at least 4 and every off-diagonal non-zero entry of is at least 3. No condition on . In this work we solve the word problem for such , show that is torsion free and show that has property , provided that has these properties, correspondingly. We also compute the homology and cohomology of , relying on that of . The two main tools used are Howie diagrams corresponding to relative presentations of with respect to and small cancellation theory with mixed small cancellation conditions.

Cite this article

Arye Juhász, Relatively extra-large Artin groups. Groups Geom. Dyn. 12 (2018), no. 4, pp. 1343–1370

DOI 10.4171/GGD/471