I am trying to learn about arithmetic Fuchsian groups from Svetlana Katok's book Fuchsian groups.
At the beginning of Chapter 5, it says that the Fuchsian group is arithmetic if it has integer coefficients. Also when I search, a more precise definition is as the following (https://www.mathi.uni-heidelberg.de/~alessandrini/Arith_Reports/6-Arithmetic%20Groups%20from%20Quaternions.pdf):
Let $T:PSL(2,\mathbb{R})\to GL(n,\mathbb{R})$ be a finite dimensional linear representation. Then the subgroup $T(PSL(2,\mathbb{R}))\cap GL(n,\mathbb{Z})$ lifts to a discrete subgroup $\Gamma:= T^{-1}(T(PSL(2,\mathbb{R}))\cap GL(n,\mathbb{Z}))$ of $PSL(2,\mathbb{R})$. All subgroups of $PSL(2,\mathbb{R})$ obtained in this manner and all their subgroups of finite index are called arithmetic Fuchsian groups.
Another definition arising from a quaternion algebra is kind of long and I will omit here, but it is in the above book and the note that I mentioned.
It is not at all clear for me whether the two definitions are equivalent. Along reading, I guess I could make the group of unit quaternions act on the quaternion algebra by left multiplications, which might give a matrix of $GL(4,\mathbb{Z})$.
So my question is, are the two definitions equivalent? If so, how can I see it? Or could you point me to a reference?
Thanks!