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Let $S_g$ be a closed orientable surface of genus $g>1$.

How can one prove that its mapping class group $\mathrm{Mod}(S_g)$ is not generated by two Dehn twists?

A pair of simple closed curves in $S_g$ may be very complicated (e.g. there are filling pairs of curves, such that they are in minimal position and they divide $S_g$ into discs). Maybe the question is still open, is it?

I also know several results in this direction, such as:

  • $\mathrm{Mod}(S_g)$ can be generated by two elements -- a product of $2$ Dehn twists and a product of $2g$ Dehn twists
    [Wajnryb, Mapping class group of a surface is generated by two elements, 1996]

  • $\mathrm{Mod}(S_g)$ can be generated by two elements, one of which is a Dehn twist
    [Korkmaz. Generating the surface mapping class group by two elements, 2004]

  • $\mathrm{Mod}(S_g)$ can be generated by two torsion elements, of order $g$ if $g\ge6$
    [Yildiz. Generating mapping class group by two torsion elements, 2020]

Maybe you can share some similar interesting results?

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2 Answers 2

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In

Humphries, Stephen P. Generators for the mapping class group. Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722, Springer, Berlin, 1979.

Humphries proves that no collection of less than or equal to 2g Dehn twists generates the mapping class group of a closed genus g surface, but that a specific collection of 2g+1 twists does generate it.

By the way, it is easy to see that no collection of at most 2g-1 twists generates the mapping class group since such a collection of twists will always fix some nonzero vector in first homology (the point being that the first homology group has rank 2g, and a Dehn twists acts trivially on a codimension one subgroup of homology).

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This is addressed in §3.5.2 of Farb and Margalit's Primer on Mapping Class Groups. The subgroups of mapping class groups generated by two Dehn twists $T_a,T_b$ are one of:

  • $\mathbb{Z}$ if the curves $a$ and $b$ are isotopic;
  • $\mathbb{Z}^2$ if $a$ and $b$ are non-isotopic and disjoint;
  • the 3-strand braid group if $i(a,b)=1$;
  • free of rank 2 otherwise.

In particular, in no case do these generate the whole mapping class group of a closed, hyperbolic surface.

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