Moduli space ($\mathcal{M}$)

A space parametrizing isomorphism classes of geometric objects. Its topology and possible stabilizers are part of the definition; a moduli space of Riemann surfaces forgets the marking retained by Teichmüller space.

The punctured torus retracts onto two circles, so its flat $SU(2)$ gauge-orbit space is $SU(2{)}^2/SU(2)$ under simultaneous conjugation. Write unit quaternions as $A=a+u$ and $B=b+v$. Complete orbit invariants are $(a,b,u\cdot v)$ with $(u\cdot v{)}^2\le (1-a^2)(1-b^2)$. Adding $ab$ to the third coordinate identifies this space with the compact convex set of positive-semidefinite $3\times 3$ real correlation matrices, a closed three-ball. The dense open subset $A\ne \pm 1$ is a closed disk times an open interval; its complement is two disjoint intervals.

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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.