Moduli space of Riemann surfaces (${\mathcal{M}}_g$)

The moduli space of unmarked compact genus-$g$ Riemann surfaces. For $g\ge 2$ it is the quotient ${\mathcal{T}}_g/\mathrm{Mod}(S_g)$ of Teichmüller space by the mapping class group, with its quotient topology. The thick subsets are compact by Mumford's compactness theorem.

For stable compact Riemann surfaces with $n$ distinct labelled marked points, where $2g-2+n>0$, the displayed complex dimension counts independent conformal shapes and positions after conformal automorphisms are removed. On the sphere it is $n-3$, since three points fix a Möbius transformation. On a torus it is $1+n-1=n$: one shape modulus and $n$ positions, less one translation. For genus $g\ge 2$, the unmarked shape dimension is $3g-3$ and continuous conformal automorphisms vanish. The shape dimension also follows from the Riemann-Roch theorem applied to holomorphic quadratic differentials.

The subset of the unmarked moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon >0$ is compact for fixed genus $g\ge 2$. Use a Bers pants decomposition theorem bound, finitely many topological decomposition types and Dehn twists to place representatives in finitely many compact boxes in Fenchel–Nielsen coordinates.