Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\rightarrow\Gamma_n$ such that for all $\sigma\in\Gamma,\tau_n(\sigma)$ converge to $\sigma$.

Do someone have an idea or a reference about the proof of the following Lemma:

**Lemma:**

Let $(\Gamma_n)_n$ be a sequence of Fuchsian groups converging to a group $\Gamma$ and $\tau_n$ the isomorphisms given in the definition above then:

There exist homeomorphisms $\phi_n:\mathbb{H}^2/\Gamma\longrightarrow\mathbb{H}^2/\Gamma_n$ whose lifts $\tilde{\phi}_n$ satisfies for any $\sigma\in\Gamma$

$$\tilde{\phi}_n\circ\sigma=\tau_n(\sigma)\circ\tilde{\phi}_n$$

and such that $(\tilde{\phi}_n)_n$ converges to the identity map uniformly on compact sets i.e that $(\tilde{\phi}_n)$ satisfies, $$\forall x\in\mathbb{H}^2, \tilde{\phi}_n(x)\underset{n\rightarrow\infty}{\longrightarrow}x$$

($\mathbb{H}^2$ is the hyperbolic plane.)

Thanks for any answer