Proof of Lemma

For 0 < r < 1, tεH(2) maps the circle Cr = {z : |z| =r}
to the locus {z : |z - t(0)|/|t(0)*z - 1| = r}.

Let t(0) = γ. Then t-1 maps γ to 0.

Suppose first that t (and hence t-1) is direct.
As t-1 maps γ to 0, the hyperbolic group page
shows that t-1(z) = κ(z-γ)/(γ*z-1).
z ε t(Cr)   if and only if   t-1(z) ε Cr,
i.e. |t-1(z)| = r.
i.e. |κ(z-γ)/(γ*z-1)| = r,
i.e. |z-γ|/|γ*z-1| = r,   as |κ| = 1.
The result follows as γ = t(0).

Now suppose that t is indirect.
As t-1is now indirect, the argument above shows that
z ε t(Cr) if and only if |(κ(z-γ)/(γ*z-1))*| = r.
But |(κ(z-γ)/(γ*z-1))* = |κ(z-γ)/(γ*z-1)|.
Thus, we have the result in this case also.

return to hyperbolic circles