It may be that I am missing something very simple. In S. Katok’s book “Fuchsian Groups”, Lemma 5.3.3, we have the following.

Lemma: Let $ \Gamma$ be a Fuchsian group of finite covolume, $ k_0=\mathbb{Q}(\mathrm{tr}(\Gamma))$ . Assume that $ [k_0:\mathbb{Q}]<\infty$ and $ \mathrm{tr}(\Gamma)\subset \mathcal{O}_{k_0}$ (the ring of integers of $ k_0$ ). Then, $ $ \mathcal{O}_{k_0}[\Gamma]=\left\{\sum_i a_i \gamma_i\,:\, a_i\in\mathcal{O}_{k_0},\,\gamma_0\in\Gamma\right\}$ $ is an order of the quaternion algebra $ $ k_0[\Gamma]=\left\{\sum_i a_i\gamma_i\,:\, a_i\in k_0,\,\gamma_0\in\Gamma\right\}. $ $

In the proof, we have the following argument.

We may assume that $ \gamma_0=\begin{pmatrix}\lambda\&\lambda^{-1}\end{pmatrix}$ , $ \lambda\neq 1$ and $ \Gamma\subseteq \mathrm{PSL}(2,K_0)$ where $ K_0=k_0(\lambda)$ . If $ \gamma=\begin{pmatrix}a&b\c&d\end{pmatrix}\in\mathcal{O}_{k_0}[\Gamma]$ , then $ a+d$ and $ \lambda a+\lambda^{-1}d$ are in $ \mathcal{O}_{k_0}$ . Notice that $ \lambda$ and $ \lambda^{-1}$ are units in $ K_0$ and $ \mathcal{O}_{k_0}$ is a subring of the ring of integers of $ K_0$ , so $ a$ and $ d$ are in the fractional ideal $ \frac{1}{\lambda^2-1} \mathcal{O}_{k_0}$ of $ K_0$ .

I don’t understand why $ a$ is in that fractional ideal. It seemed to me that $ $ (\lambda^2-1)a=\lambda(\lambda a+\lambda^{-1}d)-(a+d)\in (\lambda-1)\mathcal{O}_{k_0}-\mathcal{O}_{k_0}. $ $ But I don’t see why that should lie in $ \mathcal{O}_{k_0}$ . Also, why is that ideal fractional? Wouldn’t that imply that $ (\lambda^2-1)\in \mathcal{O}_{k_0}$ ?

On the author’s website, there is a mention of this Lemma in the errata: http://www.personal.psu.edu/sxk37/errata.pdf but it just says: “The end of the proof of Lemma 5.3.3 was modified according to M.Katzโs suggestion.”

Thanks for any clarification!