Teichmuller space

Teichmuller space $\mathcal{T}_g$ for genus $g$ parametrizes pairs $(C,\phi)$ of a genus $g$ Riemann surface $C$ and a homeomorphism $\phi:C \rightarrow C_{0}$ to a fixed surface of genus $g$, up to isotopy of $\phi$. This is equivalent to parametrizing Riemann surfaces $C$ with a choice of normalized generators of $\pi_{1}(C)$. It is naturally an open subset of $\mathbb{C}^{3g-3}$, homeomorphic to a ball. It admits an action of the mapping class group $\Gamma_g$ with finite stabilizers and with quotient $\mathcal{M}_g$.