TITLE: What do Weierstrass points look like?
QUESTION [55 upvotes]: As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points.  Smooth manifolds are totally homogeneous, but in the complex category you are just walking around, minding your own business, when suddenly you say to yourself: "this point feels different from all the others!"
I think I'd be happier if I could see a geometric feature which corresponds to the Weierstrass points.  So here is a more precise sub-question:  take a compact complex curve $X$ of genus $g \geq 2$, equipped with the hyperbolic metric.  Is there something special happening for this metric at the Weierstrass points?  Some kind of geodesic focusing, maybe?

REPLY [2 votes]: Associated to a dessin d'enfant (or hypermap) there is an associated complex analytic structure.
(See Grothendieck's esquisse d'un programme or 
 D. Singerman Automorphisms of maps, permutation groups and Riemann surfaces,
Bull. London Math. Soc. 8 (1976), 65-68.) 
Grothendieck observed that Belyi's theorem implies that the corresponding algebraic curve is defined over the field of algebraic numbers.
The most symmetric dessins are the regular maps, long studied by Coxeter and many others. For example, the famous Klein quartic corresponds to the well-known  map of type {3,7} of 24 heptagons on a genus 3 surface. Here the 24 Weierstrass points are the face-centres.
More generally we can ask whether vertices, face-centres or edge-centres are Weierstrass points. 
The answer is yes for genus 2, 3, 4 and with one possible exception for genus 5 too.
See D. Singerman and P. Watson, Weierstrass points on regular maps, Geom. Dedicata, 66
(1997), 69-88.