Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups

Question

Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence".

Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be 1 if $-I\in\Gamma$, and 0 otherwise. Let $d$ be its index of $\Gamma$ inside $SL(2,\mathbb{Z})$, let $n$ be its number of cusps, and let $w_1,\ldots,w_n$ be the width of its cusps. By the signature of $\Gamma$, I mean the tuple $(c_2,c_3,c_{-1},d,n,\{w_1,\ldots,w_n\})$.

Now suppose $\Gamma_1,\Gamma_2$ are two such finite index subgroups. Let $H$ be the upper half plane. I'd like to understand the relationship between the statements:

(A) $\Gamma_1$ is conjugate to $\Gamma_2$ in $SL_2(\mathbb{Z}$).

(A') The images of $\Gamma_1,\Gamma_2$ in $PSL_2(\mathbb{Z})$ are conjugate.

(B) $\Gamma_1$ and $\Gamma_2$ have the same signature.

(C) $H/\Gamma_1$ is isomorphic to $H/\Gamma_2$ as Riemann surfaces.

(C') $H/\Gamma_1$ is isomorphic to $H/\Gamma_2$ as complex orbifolds.

(D) $H/\Gamma_1$ is isomorphic to $H/\Gamma_2$ as Riemann surfaces over $H/SL(2,\mathbb{Z})$.

(D') $[H/\Gamma_1]$ is isomorphic to $[H/\Gamma_2]$ as complex orbifolds over the orbifold quotient $[H/SL(2,\mathbb{Z})]$.

Certainly (A) implies everything else, and is equivalent to (D').

Certainly (D') $\Rightarrow$ (D), (C') $\Rightarrow$ (C), (D) $\Rightarrow$ (C) and (D') $\Rightarrow$ (C').

Also I've computed some explicit examples showing that (B) $\not\Rightarrow$ (A).

There are some other obvious implications, but mostly other than this I know very little. In particular, what does knowing the signature really tell us? What is the relationship between (B) and any of the other properties?

Does assuming (B) change any of the relationships? For example, certainly neither (C) nor (C') imply (D) or (D'), but does (B)+(C) or (B)+(C') imply (D) or (D')?

Answer 1

First I have two comments: I am a bit surprised that you say that (D') $\Rightarrow$ (A): I believe the action on the Poincaré half-plane sees only the image in $\mathrm{PSL}_2(\mathbb Z)$. Also, your notion of "signature" is highly nonstandard: in general the width of the cusps (by which I guess you mean the degree of the covering map to the single cusp of the modular surface) is not included. You might want to call it "augmented signature" or "covering signature" or maybe something more intelligent than those two names.

Now for my (partial) answers to your questions. I am not too familiar with the complex theory so assume everything I say is in the category of hyperbolic orbifolds. In particular I will ignore (C) and (D).

(i) Note that (A) and (A') are not equivalent in general: a dumb example is when $\Gamma_1$ with $c_{-1}=1$ and $\Gamma_2$ with $c_{-1}=0$ map to the same subgroup $\Gamma$ of $\mathrm{PSL}_2(\mathbb Z)$. To be more precise, this will work exactly when such a $\Gamma_2$ exists (since $\Gamma_1$ is necessarily the full preimage in $\mathrm{SL}_2(\mathbb Z)$), which happens exactly when $\Gamma$ does not have $2$-torsion (when $\Gamma$ is torsion-free it is in fact free and this is trivial, in general this is a theorem of Kra). In this situation there are actually many lifts of $\Gamma$ (i.e. subgroups with $c_{-1}=0$ mapping to $\Gamma$): they are indexed by $H^1(\Gamma, \mathbb Z/2)$) and they are not conjugated to each other. Thus the better condition to relate to the others is probably (A') (see also my comment in the first paragraph).

(ii) A way to see that (B) and (A') are not equivalent is to prove that there are many more finite covers of a given degree than "augmented signatures" corresponding to this degree (via the covolume formula for discrete groups). Indeed, the former is factorial in the degree (this is not obvious, see section 5.2 in http://arxiv.org/abs/0811.2482 for a more general result), and if I'm not mistaken the latter should be subexponential in the degree (with 'classical' signatures, which do not contain the width of the cusps, this is true, and the width parameters multiply by a subexponential contribution since they are a partition of the degree).

In fact the argument in the paper I linked also implies that condition (B) together with the knowledge of $c_{-1}$ cannot imply either (C') or (D'), since they prove that there are factorially many of non-isometric covers of the modular surface in each of those.

(iv) To sum up, if everything I said is correct then we get the following additions to your implications:

(A) $\Rightarrow$ (A') $\Leftrightarrow$ (D') $\Rightarrow$ (C'),(B)

and the reverse implications are false in general.