The Signatures and Boundary Components of the Groups $\hat{\Gamma}_{0,n}(N)$

Abstract

In this paper, we established the group $\hat{\Gamma}_{0,n}(N)$ by group $\Gamma_{0,n}(N)$ extending with reflection. Then, we obtain boundary components in signature of the group and we get some calculation for link periods $2, 3, \infty$. And then, we constitute chain of reflections with fixed points via extended Hoore-Uzzell Theorem in the group. Finally, The number of boundary components in the signature of some groups $\hat{\Gamma}_{0,p}(p)$ and $\hat{\Gamma}_{0,p}(p^2), p$ is a prime number, and the number of link periods were found.