We study finitely generated discrete groups of hyperbolic plane isometries (including those which reverse orientation) with non-compact orbit space. A presentation by generators and relations this type group is obtained. To that end, we use the geometrical properties a fundamental region canonical form, apply Macbeath’s classical theorem on presentations simply connected spaces. Our main result...

Let Fn be the free group on n generators, and let PΣn be the group of automorphisms of Fn that send each generator to a conjugate of itself. The kernel Kn of the homomorphism PΣn → PΣn−1, induced by mapping one of the free group generators to the identity, is finitely generated. We show that Kn has cohomological dimension n − 1, and that Hi(Kn;Z) is not finitely generated for 2 ≤ i ≤ n − 1. It ...