Teichmuller space

Teichmuller space ${\mathcal{T}}_g$ for genus $g$ parametrizes pairs $(C,\varphi )$ of a genus $g$ Riemann surface $C$ and a homeomorphism $\varphi :C\to C_0$ to a fixed surface of genus $g$, up to isotopy of $\varphi$. 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 ${\mathcal{M}}_g$.