# Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action

Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $g$ surface. The famous $\mathrm{SL}_2\mathbb{R}$ action on $\Omega\mathcal{T}_g$ is defined by composing each coordinate chart of $(X,\omega)$ with matrices in $\mathrm{SL}_2\mathbb{R}$. It is claimed that by descending to $\mathbb{H}$ and $\mathcal{T}_g$ from their tangent/cotangent bundle, the $\mathrm{SL}_2\mathbb{R}$ action embeds the hyperbolic plane $\mathbb{H}$ isometrically into Teichmuller space $\mathcal{T}_g$ of genus $g$.

A few things I have learned so far:

• $\mathrm{PSL}_2\mathbb{R}$ is identified with $T^1\mathbb{H}$ by choosing $(i,i)\in T^1\mathbb{H}$ for identity matrix. For any $g\in \mathrm{PSL}_2\mathbb{R}$ identified with $(x,v)\in T^1\mathbb{H}$, $ga_t$ traces the geodesic in $\mathbb{H}$ passing through $x$ with tangent vector $v$.
• Let $a_t:=\begin{pmatrix}e^{t/2}&0\\ 0&e^{-t/2} \end{pmatrix}, t\in\mathbb{R}$. The action of $a_t$ streches the horizontal direction and shrinks the vertical direction of $(X,\omega)$. By the Teichmuller theorem, $a_t\cdot (X,\omega)$ traces the geodesic in $\mathcal{T_g}$ passing through $X$ with marking $id_X$ as $t$ varies.

So if the embedding is given by the identification of $T^1\mathbb{H}$ with $\mathrm{PSL}_2\mathbb{R}\cdot (X_0,\omega_0)$ above, the curve $t\mapsto a_tg$ in $T^1\mathbb{H}$ is sent to the geodesic $t\mapsto a_t\cdot (X,\omega)$ in $\mathcal{T}_g$ given that $(X,\omega)=g\cdot (X_0,\omega_0)$. However, the former curve $t\mapsto a_tg$ is NOT the geodesic but a scaling of the complex number $x$ to which $g$ is identified with. The fact that a non-geodesic is sent to a geodesic contradicts the claim that the embedding of $\mathbb{H}$ is isometric. I think the issue here is that the geodesic flow on $\mathrm{PSL}_2\mathbb{R}$ is a RIGHT multiplication by $a_t$ while the $\mathrm{PSL_2}\mathbb{R}$ action on $\Omega\mathcal{T}_g$ is from left.

Where does my argument go wrong? Or do we define the identification of $\mathbb{H}$ with $\mathrm{SO(2)}\backslash \mathrm{SL}_2\mathbb{R}\cdot (X_0,\omega_0)$ differently?

I apologize in advance if this kind of questions don't belong here. The link to the same post on MSE https://math.stackexchange.com/questions/2822006/teichmuller-disk-and-mathrmsl-2-mathbbr-action

• What do you call Teichmüller disk? isn't it Teichmüller space vs Poincaré disk? – YCor Jun 17 '18 at 14:02
• @YCor By Teichmueller disk I mean the isometrically embedded Poincare disk in Teichmuller space. – Morty Jun 17 '18 at 14:08
• There is now an answer to the stack-exchange version of this question. – Lee Mosher Jun 17 '18 at 16:35
• @LeeMosher Thank you for update. I have read the answer given on the other site. It is a good expansion of content of my first item in the list of this post. Like I commented below that answer, it didn't answer my question raised later in the post. – Morty Jun 17 '18 at 17:52
• If you post a question on MSE, you should wait a few days without getting answers before reposting it here, and it's common courtesy to include links between the versions of the posts so that there is no duplication of effort. – j.c. Jun 17 '18 at 19:47

I have found an answer to my question with which I am satisfied by studying the original paper of Veech: Teichmuller curves in moduli space, Eisenstein series adn an application to triangular billiards. He considered two actions on $\Omega \mathcal{T}_g$. One is the $PSL_2(\mathbb{R})$ action from left mentioned above, and the other is an action by $Aff^+(X,\omega)$, the group of orientation-preserving affine homeomorphisms from right. Although the Veech group $SL(X,\omega)$ could be defined either as stabilizer of $PSL_2(\mathbb{R})$ action or the image of differential $D:Aff^+(X,\omega)\rightarrow PSL_2(\mathbb{R})$, the geodesic flow on $\Omega\mathcal{T}_g$ should be really defined as the right action of $\phi_t\in Aff^+(X,\omega)$ whose derivative $D\phi_t=a_t$ instead of the left multiplication by $a_t$. The former is exactly what Veech did in his paper. However, the survey I read defined geodesic flow in the latter way and I was confused by this technical point.