non-discrete group isomomorphic to a discrete group I am trying to find an example of a discrete group of Möbius transformation that is isomorphic  (algebraically) to a non-discrete group.  
Can someone please help finding such groups.
 A: Here is the simplest example: Take an infinite cyclic group $G_1$ generated by a hyperbolic (or parabolic if you prefer) isometry of the hyperbolic plane. This group is discrete. Now, take an infinite cyclic group $G_2$ generated by an irrational rotation (an elliptic isometry of infinite order). Clearly, the second group is not discrete as it is dense in a subgroup of $PSL(2,R)$ isomorphic to $S^1$. If you want to see more interesting examples, read Steve D's comments. Furthermore, there are nondiscrete subgroups of $PSL(2, R)$ isomorphic to fundamental groups of closed hyperbolic surfaces. Even more:
Every finitely generated discrete infinite subgroup $G$ of $PSL(2,R)$ is isomorphic to a nondiscrete subgroup, except for a finite number of Van Dyck groups (I can list the exceptions if you are interested). 
Edit. Here is how to construct explicit examples of nondiscrete groups of surface group embeddings. Start for instance with the Coxeter triangle group $W$ whose Coxeter graph is the triangle with the edge labels $7$. This group acts as a cocompact discrete group of isometries of the hyperbolic plane with the fundamental domain which is an equilateral hyperbolic triangle $T$ with the angles $\pi/7$. Now, take an equilateral hyperbolic triangle $T'$ with the angles $2\pi/7$. It exists since $6\pi/7<\pi$. Now, take the group $W'$ generated by reflections in the edges of the new triangle. There exists an obvious homomorphism $f: W\to W'$ sending generators to generators. It requires a bit of work to show that $f$ is an isomorphism (surjectivity is clear, injectivity follows from Galois group considerations). Now, $W$ admits a finite index subgroup isomorphic to the fundamental group of closed hyperbolic surface (I can compute the genus, but it would take me some time). Restriction to this subgroup would be an explicit example, since vertices and edges of $T'$ one can compute explicitly using Lorentzian model of the hyperbolic plane. 
