Any convex cone has an accumulation point in the base by the action of its automorphism group. In this paper, we prove the converse of this statement, more precisely, a convex domain $\Omega$ with a face $F$ of codimension 1 is a cone over $F$ if there is an Aut($\Omega$)-orbit accumulating at a point of $F$.

We prove that the growth functions associated with Artin-monoids of finite type are rational functions whose numerators is equal to 1. We give an explicit formula for the denominator polynomial $N_{M}(t)$ and give three conjectures on it: 1. $N_{M}(t)$ is irreducible up to a factor 1-t, 2. there are $l$-1 real distinct roots of $N_{M}(t)$ on the interval (0,1), and 3. the smallest real root on (0,1) is the unique smallest absolute values of all roots of $N_{M}(t)$.

Let $G$ be a finite subgroup of the mapping class group of genus $\sigma$, which acts on a compact Riemann surface of genus $\sigma$. In this paper, we introduce a new method to determine the rotation angle of an element $g\in G$ around the fixed points of $g$. Our main result is Theorem 3.2.