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Some general remarks on your question: there is an algorithm which, for any $g$, will return a list of all automorphisms of a genus $g$ surface. I'll try to give one possible interpretation of what "l …

(Deleted incorrect suggestion). I think one can use uniformization and the construction of automorphic functions on the universal cover to produce meromorphic functions. A google search for these ter …

If your compact Riemann surfaces $M$ and $N$ have a hyperbolic metric in which the boundary curves are totally geodesic of the same length, then this follows from the Fenchel-Nielsen coordinate parame …

The braid groups are good (which are mapping class groups of punctured disks) by Proposition 3.5 of Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A., Cohomological goodness and the profinite com …

If the surface is closed of genus $g$, there is a universal lower bound on $\lambda_{2g-2}$.

For a), when $\chi(F)<0$ $F$-bundles over $B$ are classified by maps $\pi_1(B)\to Mod(F)$, where $Mod(F)$ is the mapping class group of $F$. This boils down to the fact that $Diff_0(F)$ is contractibl …

If you have a compact hyperbolic surface with geodesic boundary $\Sigma$, then you may double the surface along its boundary to get a closed hyperbolic surface $D\Sigma=\Sigma\cup_{\partial\Sigma}\Sig …

By the uniformization theorem, there is a unique conformally equivalent constant curvature metric up to scaling which is equivalent to a Riemann surface. A conformal Killing vector field would give a …

If the points are (setwise) invariant under reflection through a plane perpendicular to the sphere, then the the great circle of the reflection plane intersecting the sphere will be geodesic in the hy …

The action is discrete if $X$ is hyperbolic and is not a disk or annulus. By uniformization, $X=\mathbb{H}^2/\Gamma$ for some discrete subgroup $\Gamma< PSL_2(\mathbb{R})$. Let $\Lambda < PSL_2(\mat …

One may define a canonical round metric conformally equivalent to $m$ in the following way: the pullback $f^*(dvol_m)$ of the area measure of $m$ gives a finite measure on $S^2$. Regarding $S^2$ as th …

There's a remarkable theorem of Margulis that pertains to your question. Let $G$ be a semisimple Lie group (in your case, $PSL_2(\mathbb{R})$), and let $\Gamma$ be an irreducible lattice in $G$. The c …

A result of Behrstock and Minsky (cf. Hamenstadt too) implies that the rank of mapping class groups is the maximal rank of abelian subgroups, which is $3g+p-3$ for a connected hyperbolic surface of ge …

There's a slight issue I believe with the other answers. If we consider moduli space as an orbifold (of complex dimension $3g-3$), and the hyperelliptic locus an immersed suborbifold (of complex dimen …

I think that $A=2d$ will work, basically by applying Morse theory to the distance function from $x$. Morse theory for distance functions was originally considered by Gromov (and then Cheeger). For the …