Relatively extra-large Artin groups

Abstract

Let $n\geq 2$ be an integer and let $N$ be an $n \times n$ symmetric matrix with 1's on the main diagonal and natural numbers $n\_{ij}\neq 1$ as off-diagonal entries. (0 is a natural number). Let $X={x\_1,\ldots ,x\_n}$ and let $F$ be the free group on $X$. For every non-zero off-diagonal entry $n\_{ij}$ of $N$ define a word $R\_{ij}:=UV^{-1}$ in $F$, where $U$ is the initial subword of $(x\_ix\_j)^{n\_{ij}}$ of length $n\_{ij}$ and $V$ is the initial subword of $(x\_jx\_i)^{n\_{ij}}$ of length $n\_{ij}$, $1\leq i,j \leq n$. Let $A$ be the group given by the presentation $\langle X\mid R\_{ij},,n\_{ij}\geq 2 \rangle$. $A$ is called the Artin group defined by $N$, with standard generators $X$. Let $Y={x\_1,\ldots ,x\_k}, k\<n$ and let $N\_Y$ be the submatrix of $N$ corresponding to $Y$. Let $H=\langle Y \rangle$. 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.