Talk:Mapping class group of a surface

The page is in an unsatisfying state as I left it, some sections are stubs (for example those on cohomology, general subgroups and the historical introduction---they do not necessarily need to be much longer but they could stand to be more synthetic and comprehensive) and the article lacks a short discussion of the non-orientable case.

One thing I am not much satisfied with is the lack of a section on the large scale geometry of the group. In particular, a substantiation of the heuristic that the MCG behaves in some ways like an hyperbolic group and in others like a higher-rank lattice would be nice: some are scattered around the current page but having a synthetic, short exposition of this would be much better. I am not unfortunately not aware of any wide-ranging book or survey about it.

Links to relevant algebro-geometric topics would also be welcome. jraimbau (talk) 16:38, 19 July 2016 (UTC)

The Birman exact sequence
I tried to make what I thought was an improvement to the section on the Birman exact sequence. To get it right, my reference is these lecture notes by (Dylan) Thurston. I don’t know how to make that into a decent Wiki link. Unfortunately, that reference only covers $$\chi(S)< 0$$, and I’m definitely confused about something when $$S=S^2$$. I’m working on it. In the meantime, I’ve reverted by own revisions. Adam1729 (talk) 22:02, 14 October 2020 (UTC)

Contradictory passage
This passage appears in the article:

"This action thus gives a linear representation $$\mathrm{Mod}(S) \to \mathrm{GL}_{2g}(\mathbb Z)$$."

This map is in fact a surjection with image equal to the integer points $$\mathrm{Sp}_{2g}(\mathbb Z)$$ of the symplectic group."

Of course the map is a surjection onto its image. But the map referred to in "This map is in fact a surjection" is in fact not a surjection (for g ≥ 2), because its codomain is $$\mathrm{GL}_{2g}(\mathbb Z)$$ (and not $$\mathrm{Sp}_{2g}(\mathbb Z)$$).47.44.96.195 (talk) 22:11, 20 October 2020 (UTC)


 * this was indeed carelessly written (maybe by me), please edit to give it the correct meaning. jraimbau (talk) 11:25, 21 October 2020 (UTC)