TITLE: Failure of Mostow rigidity in dimension 2
QUESTION [19 upvotes]: I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question:
(1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ to itself such that for the continuation $\partial f$ of $f$ to the boundary $\partial \mathbb H^2$, there does not exist an isometry $\varphi$ of $\mathbb H$ such that $\bar{\varphi} |_{\partial \mathbb H^2} = \partial f$?
The reason I am asking this question is that one way to prove classical Mostow rigidity (two closed hyperbolic manifolds of the same dimension $n \geq 3$ with isomorphic fundamental groups are isomorphic) can be deduced from the statement that for $n \geq 3$, every quasiisometry $f$ of hyperbolic space $\mathbb H^n$ extends to a homeomorphism $\partial f: \partial \mathbb H^n \rightarrow \partial \mathbb H^n$ which is the restriction of an isometry of $\mathbb H^n$. 
The proof of this statement decomposes into two pieces: First, one shows that $\partial f$ is a homeomorphism, secondly, one proves that it is conformal and hence has to come from an isometry of $\mathbb  H^n$. The first step also works in dimension 2, but the second does not. So more general than the above, we could even ask the following (by identifying $\partial \mathbb H^2$ with $S^1$):
(2) Which self-homeomorphisms of $S^1$ arise as extensions of quasiisometries $\mathbb H^2 \rightarrow \mathbb H^2$ to the boundary?

REPLY [6 votes]: The actual questions you asked have already been answered, and so I am hesitant to chime in with anything.
Anyway, here is an different response to your opening line: 

"I am trying to understand why Mostow rigidity fails in dimension 2."

Here is my first thought:

The structure of the fundamental group of surfaces allows for local deformations in the moduli space of hyperbolic structures.

For every hyperbolic structure on a closed $n$-manifold there is a discrete faithful representation (hence irreducible) from $\pi_1(M)\to \mathrm{Isom}(M)$, unique up to conjugacy.
When $M$ is a surface, $\mathrm{Isom}(M)$ is $\mathrm{PSL}(2,\mathbb{R})$ and $\pi_1(M)$ has a presentation with $2g$ generators and 1 non-trivial relation in those generators.  One can use this to conclude that: $\mathrm{Hom}(\pi_1(M),\mathrm{Isom}(M))/\mathrm{Isom}(M)$ has dimension $3(2g-1)-3=6g-6$; and locally so at any irreducible representation.
So there are non-trivial infinitesimal deformations in that case.  However, the infinitesimal deformations can be "integrated", since the obstructions are in second cohomology which vanishes since $n=2$.  Hence there is no rigidity (local and hence not global).
For $n>2$, there are no such deformations; which follows from Weil's Remarks on the cohomology of groups (see here) also using information about a presentation of $\pi_1(M)$.  One concludes that there are no infinitesimal deformations and thus no local deformations. 
Mostow's Rigidity says there is global rigidity (for $n>2$); admittedly a stronger statement than local rigidity (although infinitesimal rigidity is stronger than local rigidity).  But the difference between $n=2$ and $n>2$ is seen locally as well and mostly is a consequence of nature of $\pi_1(M)$ in those cases.
So one can say:

Mostow Rigidity Fails in dimension 2 since there is no local rigidity in dimension 2, which follows mostly from the structure of $\pi_1(M)$.