LInearity of Tubular Groups

We prove that a tubular group is linear if and only if it is residually finite, equivalently virtually primitive. Every such group embeds in ${\text{SL}}_n(\mathbb{Z}[u])$. For a primitive tubular decomposition with at most $m$ edge slopes at each vertex, our representation has dimension $2\max \{2,m\}$. In particular, Gersten’s group embeds in ${\text{SL}}_6(\mathbb{Z}[u])$, and this dimension is minimal.