# Explicit description (=pictures!) of elements in $Mod_g[k]$?

This question is probably obvious to experts but I couldn't find the answer in the literature...

Background: Consider the mapping class group $Mod_g$ of the closed genus $g$ surface. There are many nice sets of generators (i.e. Humpreys famous $2g+1$ Dehn twists or Wajnryb's 2-element generating set etc).

If we consider the Torelli group $\mathcal{I}_g = Mod_g[1]$, we know it is finitely generated by bounding pair maps. D. Johnson proved that the next level in the Johnson filtration $Mod_g[2]$ (now called the Johnson kernel) is generated by Dehn twists around separating curves.

My question:

1. Is there an explicit description of elements deeper in the mapping class group (i.e. in $Mod_g[k]$ with $k \geq 2$)? Or at least examples in picture form...
2. Is there any known characterization (even for special values of $g$ and $k$) of such elements?
3. Finally (unrealistically optimistic) - is there a known generating set?

## 1 Answer

Generators for the higher terms in the Johnson filtration are not known.

Probably the best way to find explicit elements is to use the fact that the Johnson filtration forms a central filtration, and thus the kth term of the lower central series of the Torelli group lies in the kth term of the Johnson filtration. You can thus get elements by taking iterated commutators of elements in the Torelli group (e.g. bounding pair maps or separating twists). Unfortunately, it is known that this will not give generating sets; indeed, Hain proved in

R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), no. 3, 597–651.

that the lower central series and Johnson filtrations of the Torelli group are not only different, but even define different topologies on the Torelli group.

There are some things that are known about generating sets for the Johnson filtration. Let me point out three of them.

I. In my paper

A. Putman, Small generating sets for the Torelli group, Geom. Topol. 16 (2012), no. 1, 111–125.

I construct a generating set for the Torelli group which is much smaller than Johnson's generating set (its cardinality is roughly $c g^3$ for some constant $c$, while Johnson's is exponential in the genus; the abelianization has rank on the order of $g^3$, so this is the best you can do).

II. In our paper

T. Church & A. Putman, Generating the Johnson filtration, Geom. Topol. 19 (2015), no. 4, 2217–2255.

we prove that for all $k$, there exists some $G_k$ such that the kth term of the Johnson filtration is generated by elements supported on subsurfaces of genus at most $G_k$.

III. Very recently Ershov-He proved that the Johnson kernel subgroup is finitely generated for $g \geq 5$. The was extended independently by Ershov-He and by Church-Putman to prove that the kth term of the Johnson filtration is finitely generated as long as the genus is sufficiently large. The relevant papers are

M. Ershov & S. He, On finiteness properties of the Johnson filtrations, preprint 2017

and

T. Church & A. Putman, Generating the Johnson filtration II: finite generation, preprint 2017.

These can be downloaded from Ershov's webpage here and from my webpage here. The other papers by myself that I list above can also be downloaded from my webpage.

• Wow, that's a very extensive answer :-) , thanks Andy! – Nati Apr 16 '17 at 1:10
• I have one additional question: you mentioned "best way to find explicit elements" is by taking iterated commutators of BP's or separating twists. That's actually the only way I'm aware of ... is there another explicit construction? – Nati Apr 16 '17 at 3:53
• I don't know any other systematic way to find elements, though it is not hard to find various ad-hoc constructions of elements that are at least not obviously in the kth term of the lcs. – Andy Putman Apr 16 '17 at 4:21