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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 …

I think Scott's argument is that the lengths of $6g-6$ curves can't form coordinates for Teichmuller space. If one has $6g-6$ geodesics which parameterize, then they must be filling (they meet every s …

I wondered this myself, I made some similar pictures to approximate the stable lamination of a pseudo-Anosov map (the blue curve is a geodesic, the other colors horocycles). Every geodesic on the mod …

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 …

For a hyperbolic element $A\in SL(2,\mathbb{Z})$, we have the length of the closed geodesic is given by $\ln[(tr^2(A)-2+\sqrt{tr^4(A)-4tr^2(A)})/4]$, and this is monotonic in $|tr(A)|$ for $|tr(A)|>2$ …

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 …

You can have a sequence surfaces $X_i$ with arbitrarily short geodesics but $\lambda_{X_i}$ bounded away from zero. The geodesics are non-separating, so when they shrink to length zero, the Cheeger co …

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 …

A compact Riemann surface of genus $g$ with $n$ boundary components has a unique realization as a hyperbolic surface with geodesic boundary. One may see this by reflecting through the boundary and uni …

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 …

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 …

I think the short answer to your question is no, since the notion of a holomorphic map is purely local, whereas the existence of a hyperbolic metric (uniformization) depends on the global structure of …

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 …

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

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 …