Abelianizations of arithmetic Fuchsian groups

Question

Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group \begin{equation*} \Gamma=\left\{\begin{bmatrix} k+\sqrt{a}l & m+\sqrt{a}n \\ b(m-\sqrt{a}n) & k-\sqrt{a}l \\ \end{bmatrix} \colon k,l,m,n\in\mathbb Z, k^2-al^2-bm^2+abn^2=1\right\}. \end{equation*} I would like to know a reference where the abelianization of $\Gamma$ is described. Can one find an explicit non-zero group homomorphism $\Gamma\to\mathbb Z$?

Answer 1

Let $\mathbb H^2$ be the hyperbolic plane, on which $\Gamma$ acts properly discontinuously and cocompactly. Thus the quotient space $O = \Gamma \backslash \mathbb H^2$ is a closed hyperbolic surface with conical singularities. If these have angles $2\pi/q_1, \ldots, 2\pi/q_s$ then a presentation for $\Gamma$ is given by $$ \Gamma \cong \langle a_1,b_1 \ldots, a_g,b_g, c_1, \ldots, c_s | [a_1, b_1] \cdots [a_g, b_g] \cdot c_1 \cdots c_s,\, c_1^{q_1}, \ldots, c_s^{q_s} \rangle$$ from which the abelianisation is immediate to read.

There is no closed formula for the invariants $g, q_1, \ldots, q_s$. They can be computed in a given example by using software developed by John Voight (see the answer to this related question : How do you find the genus of a Fuchsian group derived from a quaternion algebra?).

There is also an asymptotic relation between $g$ and the covolume of $\Gamma$. Namely, if $\mathrm{Vol}(O)$ is the hyperbolic volume of $O$ then it holds that $$ g \sim \frac{\mathrm{Vol}(O)}{4\pi},\: s, q_i = o(\mathrm{Vol}(O)$$ as $\mathrm{Vol}(O) \to +\infty$ (this follows from work of Mikołaj Frączyk, see https://arxiv.org/abs/1810.10515 ; you could even get effective bounds by being more careful). In particular this recovers the result of Long--Maclachlan--Reid in Ian Agol's comment that the abelianisation is infinite for almost all $\Gamma$, since the volume is a proper function of $(a, b)$ (there is a closed formula but you need first to compute the discriminant of the quaternion algebra from the Hilbert symbol, which at this moment i'm not exactly sure how to do in general).

I'm not aware of any general construction of a non-trivial morphism from $\Gamma$ to $\mathbb Z$. Maybe the Jacquet--Langlands correspondence could be used for that, though i'm not sure how explicit this would be.