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I'm interested in what is known/expected about some families of arithmetic Fuchsian groups. Here is the simplest family that I'm interested in: Let $E = Z[\omega]$, where $\omega = e^{2 \pi i / 3}$. Consider the family of $Z$-valued binary Hermitian forms on $E$: $$H_n(x,y) = x \bar x - n y \bar y, \text{ for all } x,y \in E.$$ Here, we let $n$ range over all positive integers which are not norms from $E$, i.e., all positive integers for which $H_n$ does not nontrivially represent zero.

Let $\Gamma_n$ be the special unitary group of $H_n$, i.e., $$\Gamma_n = \{ g \in SL_2(E) : H_n( g(x,y)) = H_n(x,y) \text{ for all } x,y \in E \}.$$ Here $g(x,y)$ denotes the effect of matrix multiplication.

As $H_n$ is an indefinite Hermitian form, these groups $\Gamma_n$ are discrete subgroups of the Lie group $SU(1,1)$. There's almost certainly another way of seeing these groups as coming from orders in indefinite quaternion algebras over $Q$. So I guess that these groups $\Gamma_n$ yield a family of Shimura curves of increasing "complexity" measured in any natural way. Associated to the groups $\Gamma_n$ are compact orbifold Riemann surfaces $X_n$, each with invariants including genus $g$, and a series of $t$ marked orbifold points, with indices $m_1, \ldots, m_t$.

What do we know about the following: How do we expect the genus $g = g(X_n)$ to behave as $n \rightarrow \infty$? This I anticipate is known or well-studied.

But also, how does the family of indices $m_1, \ldots, m_t$ behave asymptotically? For example, how many orbifold points of index $3$ do we expect on $X_n$, as $n \rightarrow \infty$?

By some hyperbolic geometry, we can relate these indices to the volume of $X_n$. Can we use this to get some heuristics?

Am I asking something silly? I know that one can eliminate torsion by passing to a finite-index subgroup, but I would anticipate that torsion doesn't disappear in a family of groups such as the $\Gamma_n$.

References and speculations are welcome!

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I think this paper might be relevant to your question: ams.org/mathscinet-getitem?mr=1108918 –  Ian Agol Jan 6 '13 at 0:23

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