# Relatively extra-large Artin groups

### Arye Juhász

Israel Institute of Technology, Haifa, Israel

## Abstract

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 $n_{ij}=1$ as off-diagonal entries. (0 is a natural number). Let $X={x_{1},…,x_{n}}$ 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 $(x_{i}x_{j})_{n_{ij}}$ of length $n_{ij}$ and $V$ is the initial subword of $(x_{j}x_{i})_{n_{ij}}$ of length $n_{ij}$, $1≤i,j≤n$. Let $A$ be the group given by the presentation $⟨X∣R_{ij},n_{ij}≥2⟩$. $A$ 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 $k×k,k×l,l×k,l×l$, 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 $N_{Y}$. In this work we solve the word problem for such $A$, show that $A$ is torsion free and show that $A$ has property $K(π,1)$, 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.

## 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