Ytukyg

Let $Gamma$ be a congruence subgroup of $operatorname{SL}_2(mathbb Z)$. The quotient $Gamma backslash mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient map $pi: mathbb H rightarrow Gamma backslash mathbb H$ is holomorphic. There is a nice fundamental domain $D subset mathbb H$ coming from the usual fundamental domain for $operatorname{SL}_2(mathbb Z)$ which, up to some boundary identification, gives us a copy of $Gamma backslash mathbb H$ inside $mathbb H$.

The Borel measure $mu = frac{dx dy}{y^2}$ on $mathbb H$ descends to a Borel measure $overline{mu}$ on $Gamma backslash mathbb H$ which we may define using the fundamental domain: if $U subset Gamma backslash mathbb H$ is Borel, then we set

$$overline{mu}(U) := mu bigg(pi^{-1}(U) cap D bigg) tag{1}$$

Now, assume $Gamma$ is an arbitrary discrete subgroup of $operatorname{SL}_2(mathbb R)$.

1. Is there a canonical measure $bar{mu}$ on $Gamma backslash mathbb H$ coming from $mu = frac{dx dy}{y^2}$?




2. Does there always exist a fundamental domain $D$ for $Gamma$?




3. Can $overline{mu}$ arise from a differential form on $Gamma backslash mathbb H$? That is, does $overline{mu}$ come from a (unique?) smooth differential $2$-form $overline{omega}$ on $Gamma backslash mathbb H$ (thought of as a smooth manifold) which pulls back to the differential form on the smooth manifold $mathbb H$ corresponding to $mu = frac{dx dy}{y^2}$?

For intuition, I'm thinking of $mathbb R$ modulo the action of $mathbb Z$. Up to boundary identification, $[0,1]$ is a fundamental domain for the action of $mathbb Z$ on $mathbb R$. For the Haar measure $bar{mu}$ on $mathbb R/mathbb Z$, we can get it in two ways. First, if $pi: mathbb R rightarrow mathbb Z$ is the quotient map, we can measure subsets of $mathbb R/mathbb Z$ by pulling them back to $mathbb R$, intersecting them with $[0,1]$, then measuring. Second, $bar{mu}$ comes from the unique invariant nonvanishing $1$-form on $mathbb R/mathbb Z$ which pulls back to the top form $dx$ on $mathbb R$ giving the Lebesgue measure on $mathbb R$.

I'd recommend not thinking that choice of a "nice fundamental domain" is of much importance for the basic development of things. Yes, the details of a fundamental domain tell something about generators of the discrete group $Gamma$, but that's not universally necessary, nor intelligible.

So, for example, the key relationship between functions on $mathfrak H$ and $Gammabackslash mathfrak H$, or, equivalently, $Gammabackslash G$ and $G$, or $Gammabackslash G/K$, where $G=SL_2(mathbb R)$ and $K=SO(2,mathbb R)$, can be described very well without any mention or choice of "fundamental domain". This is fortunate. Namely, one proves the lemma that the averaging map $fto sum_{gammain Gamma} fcirc gamma$ from $C^o_c(G)$ to $C^o_c(Gammabackslash G)$ is surjective. Then the uniqueness of invariant distributions... shows that there is a unique integral/measure on $Gammabackslash G$ such that "unwinding" is correct, namely, that
$$
int_G f(g);dg ;=; int_{Gammabackslash G} sum_{gammainGamma} f(gamma g);ddot{g}
$$
with $ddot{g}$ denoting that measure on the quotient.

(Happily for us, the only things that this set-up depends upon are that $G$ be a unimodular topological group, and $Gamma$ a discrete subgroup.)

To answer the question asked, yes, every discrete subgroup of $SL_2(mathbb R)$ has a fundamental domain, called a Dirichlet domain.

Let $G = operatorname{SL}_2(mathbb R), K = operatorname{SO}_2(mathbb R)$, and $Gamma$ a discrete subgroup of $G$. We identify the measures $mu, dot mu, bar{mu}$ on $G, Gamma backslash G$ and $G/K$ respectively with positive linear functionals $T, dot T, bar{T}$ on $mathscr C_c(G), mathscr C_c(Gamma backslash G), mathscr C_c(G/K)$ respectively. For example,

$$T(f) = intlimits_G f(g)dg tag{$f in mathscr C_c(G)$}$$

$$bar{T}(f) = intlimits_{G/K} f(gK)dbar{mu}(gK) tag{$f in mathscr C_c(G/K)$}$$

The upper half plane $mathbb H$ with the hyperbolic measure $frac{dxdy}{y^2}$ are identified with $G/K$ and $bar{mu}$, respectively.

Since $K$ is compact, we have a natural identification of $mathscr C_c(G/K)$ with those elements of $mathscr C_c(G)$ which are right $K$-invariant, by precomposing with the quotient map $G rightarrow G/K$. We may similarly identify $mathscr C_c(Gammabackslash G/K)$ with the right $K$-invariant elements of $mathscr C_c(Gamma backslash G)$ via the quotient map $Gamma backslash G rightarrow Gamma backslash G/K$. The surjection

$$delta: mathscr C_c(G) rightarrow mathscr C_c(Gamma backslash G)$$

$$delta(f)(Gamma g) = sumlimits_{gamma in Gamma} f(gamma g)$$

can be shown to restrict to a surjection

$$mathscr C_c(G/K) rightarrow mathscr C_c(Gamma backslash G /K)$$

The answer to my question follows if I can prove the following proposition.

Proposition: There exists a unique linear positive functional $widetilde{T}$ on $mathscr C_c(Gamma backslash G/K)$, with corresponding measure $widetilde{mu}$, such that

$$bar{T} = widetilde{T} circ delta$$

In terms of measures, what this says is that if we define $widetilde{T}(f) =: intlimits_{Gamma backslash G/K} f(Gamma g K) dwidetilde{mu}(Gamma g K)$, then

$$ intlimits_{G/K} f(gK) d bar{mu}(gK) = intlimits_{Gamma backslash G/K} space space spacebigg( sumlimits_{gamma in Gamma} space space f(gamma g K) bigg)space d widetilde{mu}(Gamma gK) $$

or

$$intlimits_{mathbb H} f(z)y^{-2} dxdy = intlimits_{Gamma backslash mathbb H} sumlimits_{gamma in Gamma} f(gamma.z) d widetilde{mu}(Gamma z)$$