Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $SL_2(\mathbb{R})$. When the image $\pi(SL_2(\mathbb{R})\cdot x)\subset M_g$ of an orbit is a closed algebraic curve of $M_g$ it's called a Teichmuller curve. This is the most common definition of Teichmuller curve. On the other hand sometimes they are also referred to as affine invariant submanifolds (i.e. complex submanifolds of a stratum of $\Omega M_g$ which are $SL_2(\mathbb{R})$-invariant and linear in the period coordinates). My confusion stems from the following fact: if $(X,\omega)$ generates a Teichmuller curve then its stabiliser under the action of $SL_2(\mathbb{R})$ is a lattice and the orbit would have real dimension equal to $dim SL_2(\mathbb{R})=3$, so I don't see how to interpret a Teichmuller curve as an affine invariant submanifold.

$\DeclareMathOperator\SL{SL}$Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $\SL_2(\mathbb{R})$. When the image $\pi(\SL_2(\mathbb{R})\cdot x)\subset M_g$ of an orbit is a closed algebraic curve of $M_g$ it's called a Teichmüller curve. This is the most common definition of Teichmüller curve. On the other hand sometimes they are also referred to as affine invariant submanifolds (i.e. complex submanifolds of a stratum of $\Omega M_g$ which are $\SL_2(\mathbb{R})$-invariant and linear in the period coordinates). My confusion stems from the following fact: if $(X,\omega)$ generates a Teichmüller curve then its stabiliser under the action of $\SL_2(\mathbb{R})$ is a lattice and the orbit would have real dimension equal to $\dim \SL_2(\mathbb{R})=3$, so I don't see how to interpret a Teichmüller curve as an affine invariant submanifold.