Relatively extra-large Artin groups

Abstract

Let $n\ge 2$ be an integer and let $N$ be an $n\times n$ symmetric matrix with 1's on the main diagonal and natural numbers $n_{ij}\ne 1$ as off-diagonal entries. (0 is a natural number). Let $X=\{x_1,\dots ,x_n\}$ and let $F$ be the free group on $X$. For every non-zero off-diagonal entry $n_{ij}$ of $N$ define a word $R_{ij}:=U{V}^{-1}$ in $F$, where $U$ is the initial subword of $(x_i{x}_j{)}^{n_{ij}}$ of length $n_{ij}$ and $V$ is the initial subword of $(x_j{x}_i{)}^{n_{ij}}$ of length $n_{ij}$, $1\le i,j\le n$. Let $A$ be the group given by the presentation $⟨X\mid R_{ij},n_{ij}\ge 2⟩$. $A$ is called the Artin group defined by $N$, with standard generators $X$. Let $Y=\{x_1,\dots ,x_k\},k<n$ 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 $k\times k,k\times l,l\times k,l\times l$, 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 $N_Y$. 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.