# Convergence of Fuchsian groups and existence of suitable homeomorphisms

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.)

• I think, this was asked earlier at MSE. The answer, as before, that without further assumptions, your lemma is false. – Moishe Kohan May 9 at 20:41
• @MoisheKohan why it is false? For me the Lemma is true like that – Jongar Jongar May 9 at 20:47
• You can find a sequence of free groups of infinite rank $\Gamma_n$ converging to a free group of infinite rank $\Gamma$ such that $H^2/Gamma$ is a planar surface, while each $H^2/\Gamma_n$ has infinitely many handles. Hence, homeomorphisms $\phi_n$ do not exist. – Moishe Kohan May 9 at 21:42
• @SantanaAfton: Lemma holds for finitely generated groups, but I am not sure the result is stated anywhere in the literature. – Moishe Kohan May 10 at 14:25
• In fairness to the OP, the phrase "Fuchsian groups" is used inconsistently in the literature. Often, it includes the hypothesis of finite generation. – HJRW May 11 at 10:38