Many examples of non-cocompact Fuchsian groups sitting in PSL2(Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {PSL}_2({\mathbb {Q}})$$\end{document}

We will explicitly construct many non-arithmetic Fuchsian groups while controlling some geometric properties of the action on the boundary of the hyperbolic plane. With this construction, we produce infinitely many noncommensurable non-cocompact Fuchsian groups of finite covolume sitting in PSL2(Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {PSL}_2({\mathbb {Q}})$$\end{document} so that the set of hyperbolic fixed points of each group will contain a given finite collection of elements in the boundary of the hyperbolic plane.