Counter example to Mostow's rigidity theorem for 2-manifolds. I am trying to understand a counter-example to Mostow's rigidity theorem.
Here is the counter example I want to understand.
Take two non-isometric octagons with the sum of interior angles equal to $2\pi$. Then they form fundamental domains for non-isometric surfaces.
My questions are: how can we visualise such two hyperbolic octagons, how can we see that the fundamental groups of these surfaces would be isomorphic?
 A: Since the question itself has already been answered, this answer will add some references for further study. As Joseph Zambrano points out in the comments to his answer, "local rigidity" fails for two dimensional hyperbolic structures.  This is a hint of the deep, beautiful theory of moduli spaces of geometric structures. 
Bill Goldman's survey paper "Geometric Structures and Varieties of Representations" is dense but readable. For a deeper, foundational guide to geometric structures --- and really, if you're interested in hyperbolic geometry at all --- Thurston's notes can be found here. Chapter five is devoted to the study of the flexibility and rigidity of hyperbolic structures.
A good place to start in the literature on local rigidity is Garland's paper, which proves local subgroup rigidity under certain conditions. A good place to start on the deformation space of hyperbolic surfaces is Hubbard's Teichmuller Theory.
A: Are you familiar with the upper-half plane model of the hyperbolic plane? In this model, we consider the upper half plane (like $\mathbb{R}^2$ with y values greater than $0$); however, a geodesic ("line") between two points is either a circle centered on the x-axis or a vertical line (depending on whether the points have different x coordinates). You can construct non-isometric octagons using these geodesics with interior angle sum of $2\pi$. Now from these octagons we may construct two, non-isometric surfaces of genus $2$ (connected sum of two tori). It should not be too difficult to imagine that these surfaces will have the fundamental group as they share the same number of "holes".
