or special points. Furthermore the distinguished cycle $c_0$ is called the outside circle or the perimeter and the vertex $\del(\mk(c_0))$ is called the global zero. And the vertices $\del(\mk(c_i)),i\neq 0$ are called the local zeros. In pictures these are represented by lines rather than fat dots.

\[setrem\] It is clear that as sets $\Cacti(n)=\Cact(n)\times
(S^1)^{\times n}$ and $\cact(n)= \cact^1(n)\times
\mathbb{R}_{>0}^{\times n}$.

For the first statement one notices for each lobe $v_i$ there is a unique lowest intersection point $b$ which is the vertex of the outgoing edge of $v$. Thus there is a canonical map $\phi'_i:S^1\rightarrow |c_i|$ which starts at $b$ and goes around the cycle opposite its natural orientation. So to each cycle we associate $(\phi'_i)^{-1}(\del(\mk(c_i)))$ that is the co-ordinate of the spine as measured by $\phi'_i$. This gives the projection onto the factors $(S^1)^{\times n}$. The projection onto the first factor is given by forgetting the spines, i.e. contracting the edges $\mk(c_i)$ if $\val(\del(\mk(c_i)))=2$ and changing the marking to the unique marking which makes the graph spineless.

For the second statement the first projection is given by homogeneously scaling the weights of the edges of each non-marked cycle so that their lengths are one. The projection to the 