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Let $\mathbb{H}^2$ be the hyperbolic plane , and $(\Gamma_n)_n$ a sequence of Fuchsian groups converging to a group $\Gamma$ (i.e. there exist isomorphisms $\tau_n:\Gamma\rightarrow\Gamma_n$ such that for all $\sigma\in\Gamma,\tau_n(\sigma)$ converge to $\sigma$. )

Let $Dir_n$ and $Dir$ be the Dirichlet domain for $\Gamma_n$ and $\Gamma$ respectively. We know that the Dirichlet domain is a polygone with vertices.

My question: how to prove that the vertices of $Dir_n$ converges to the vertices of $Dir$?

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Dirichlet domains are not unique; the same Fuchsian group has many different Dirichlet domains.

Because of that nonuniqueness, the straight answer to your question is there are counterexamples: let $\Gamma_n$ to be the constant sequence converging to itself otherwise known as $\Gamma$; take each $Dir_n$ to be one fundamental domain; and I can take $Dir$ to be a different fundamental domain.

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