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For a finite index subgroup $\Gamma\le\text{SL}(2,\mathbb{Z})$, what does the coset space $\text{SL}(2,\mathbb{R})/(\Gamma\cdot\text{SO}(2,\mathbb{R})$ look like? (Here $\Gamma\cdot\text{SO}(2,\mathbb{R})$ denotes the subgroup generated by $\Gamma$ and $\text{SO}(2,\mathbb{R})$).

What is its relation with the double coset space $\Gamma\setminus SL(2,\mathbb{R})/SO(2,\mathbb{R})$ (which is homeomorphic to the "modular curve" $\Gamma\setminus\mathcal{H}$).

I'm happy to restrict to $\Gamma$ being a congruence subgroup if that is easier.

  • It is unclear to me what your quotient means since $\Gamma\cdot SO(2)$ is not a subgroup. Or maybe you mean the subgroup they generate? – Moishe Kohan Apr 24 '19 at 23:00
  • @MoisheKohan Yes, sorry I mean the subgroup they generate. –  Apr 24 '19 at 23:04
  • If $X = X. SO_2$ then $SL_2 = \bigcup_{a \in A} a X$ iff $\mathcal{H}=SL_2 .i = \bigcup_{a \in A} a X.i$, in general if $X$ is not a group there will be no $A$ for which the union is disjoint. If $X$ is the group generated by $\Gamma,SO_2$ then I would guess it is the whole of $SL_2$ because $SO_2.x.i$ is a curve in $\mathcal{H}$ – reuns Apr 24 '19 at 23:23
  • The subgroup they generate is dense but not closed, so the quotient is some awful non-Hausdorff space. What is it good for? – Moishe Kohan Apr 24 '19 at 23:30
  • @MoisheKohan Hmmm actually I think this quotient is probably trivial, if I'm understanding this correctly: https://math.stackexchange.com/questions/946148/is-there-a-group-between-so2-mathbbr-and-sl2-mathbbr

    This space came from a group action that wasn't well-defined, though I didn't realize it wasn't well-defined until literally just 10 minutes ago.

     –  Apr 24 '19 at 23:39
  • Right, the subgroup they generate is the entire SL(2,R). – Moishe Kohan Apr 24 '19 at 23:55

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