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Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

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Bounds on the uniformization map for a metric on the 2-sphere

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 …
Ian Agol's user avatar
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5 votes

Conformal Killing vector fields on compact surface of genus \ge 1

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 …
Ian Agol's user avatar
  • 68.9k
5 votes

Classification of surface bundles over surfaces

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 …
Ian Agol's user avatar
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17 votes

Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$

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 …
Ian Agol's user avatar
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3 votes
Accepted

fundamental domains in H^2 containing large balls

One can answer your question positively, although I haven't tried to compute $g_R$ as a function of $R$. One can tessellate a certain right-angled pentagon by $238$ triangles. Moreover, a genus 2 s …
Ian Agol's user avatar
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6 votes
Accepted

Simple Closed Hyperbolic Geodesics on Punctured Spheres

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 …
Ian Agol's user avatar
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21 votes

Are mapping class groups of orientable surfaces good in the sense of Serre?

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 …
Ian Agol's user avatar
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3 votes
Accepted

Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter

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 …
Ian Agol's user avatar
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20 votes
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The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be p...

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 …
Ian Agol's user avatar
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7 votes

Hyperelliptic loci in Teichmueller spaces

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 …
Ian Agol's user avatar
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5 votes
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Quasi-isometric embeddings of the mapping class group into the Teichmuller space

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 …
Ian Agol's user avatar
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6 votes

Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

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 …
Mariano Suárez-Álvarez's user avatar
2 votes

The existence of meromorphic functions on Riemann surfaces

(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 …
Ian Agol's user avatar
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3 votes
Accepted

Characterization of the moduli space of the pair of pants in terms of the modules of the ext...

I think the map $F$ is injective, but not surjective. There is a unique conformal map of the interior of the pants to the complement of 3 slits in $\mathbb{RP}^1\subset \mathbb{CP}^1$, up to the act …
Ian Agol's user avatar
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11 votes
Accepted

Growth of smallest closed geodesic in congruence subgroups?

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 $|tr(A)|>2$ …
Ian Agol's user avatar
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