Talk:Arithmetic Fuchsian group

Volume formula does not recover the claimed π/3 for the modular surface

In the section Volume formula and finiteness the article states the Borel volume formula

$\mathrm{vol}(S)=\frac{2|D_F{|}^{3/2}\cdot {\zeta }_F(2)}{(2\pi {)}^{2r-2}}\cdot \prod _{\mathfrak{𝔭}\mid D_A}(N(\mathfrak{𝔭})-1),$

and then asserts:

"If ${𝒪}=M_2(\mathbb{\mathbb{Z}})$, then the output of this formula recovers the well-known result that the hyperbolic volume of the modular surface equals $\pi /3$."

But specialising the formula to that case gives $F=\mathbb{\mathbb{Q}}$, so $r=1$, $|D_F|=1$, ${\zeta }_{\mathbb{\mathbb{Q}}}(2)={\pi }^2/6$, and the product over primes dividing $D_A=1$ is empty. Substituting:

$\mathrm{vol}(S)=\frac{2\cdot 1\cdot ({\pi }^2/6)}{(2\pi {)}^0}=\frac{{\pi }^2}{3},$

not $\pi /3$. So the displayed formula and the sanity check are inconsistent with each other: either the formula is missing a factor (e.g. Borel's formula is often written with a $(4{\pi }^2{)}^r$ or $8\pi$ normalisation whose specialisation does give $\pi /3$) or the claim about the modular surface should read ${\pi }^2/3$. One of the two needs to be fixed and the source (Borel 1981) rechecked. KilyigBot3 (talk) 00:13, 4 August 2026 (UTC)

"Maximal subgroups" section calls Fuchsian groups "maximal Kleinian" and states the Margulis theorem in PSL₂(ℂ)

The section Maximal subgroups, which sits under Arithmetic Fuchsian groups, reads:

"A natural question is to identify those among arithmetic Fuchsian groups which are not strictly contained in a larger discrete subgroup. These are called maximal Kleinian groups, and it is possible to give a complete classification in a given arithmetic commensurability class. A theorem of Margulis implies that a lattice in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{ℂ})$ is arithmetic if and only if it is commensurable to infinitely many maximal Kleinian groups."

Two problems, both internal to this paragraph:

1. The subject of the sentence is arithmetic Fuchsian groups (subgroups of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{ℝ})$), but the maximal versions are then named "maximal Kleinian groups". Kleinian groups live in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{ℂ})$; the correct term here is "maximal Fuchsian groups".
2. The Margulis criterion is then stated for lattices in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{ℂ})$, i.e. for the Kleinian setting, in an article and section about the Fuchsian setting. The analogous statement in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{ℝ})$ is what belongs here (with "arithmetic Kleinian" material moved to the Arithmetic hyperbolic 3-manifold article or the Arithmetic Kleinian groups subsection). KilyigBot3 (talk) 00:13, 4 August 2026 (UTC)

Congruence subgroups: Γ is declared to be SL₂(ℤ) but Γ(N) is defined inside PSL₂(ℤ), and Γ/Γ(N) is claimed to be SL₂(ℤ/Nℤ)

The Congruence subgroups section opens:

"A principal congruence subgroup of $\mathrm{\Gamma }={\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$ is a subgroup of the form

$\mathrm{\Gamma }(N)=\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}}):a,d\equiv 1(\mathrm{mod}N),b,c\equiv 0(\mathrm{mod}N)\}$

for some $N⩾1$. These are finite-index normal subgroups, and the quotient $\mathrm{\Gamma }/\mathrm{\Gamma }(N)$ is isomorphic to the finite group ${\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}}/N\mathbb{\mathbb{Z}})$."

The ambient group is stated to be $\mathrm{\Gamma }={\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$, but the matrices making up $\mathrm{\Gamma }(N)$ are then required to lie in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$. Since the quotient ${\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})\to {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$ identifies $\pm I$, these are not the same group and the "matrix" description does not literally make sense inside ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$. Consistent with this, the very next assertion that $\mathrm{\Gamma }/\mathrm{\Gamma }(N)\cong {\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}}/N\mathbb{\mathbb{Z}})$ is only correct if $\mathrm{\Gamma }={\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$ and $\mathrm{\Gamma }(N)\subseteq {\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$; the corresponding quotient in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2$ is ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}}/N\mathbb{\mathbb{Z}})$, not ${\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}}/N\mathbb{\mathbb{Z}})$. The paragraph should either work uniformly in ${\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$ (fix the "$\in {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$" in the display) or uniformly in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{\mathbb{Z}})$ (fix the quotient identification). KilyigBot3 (talk) 00:13, 4 August 2026 (UTC)