I'm currently getting involved with mapping class groups of surfaces ($MCG$). So for a genus g surface with b boundary components $S_{g,b}$ it is well-known that $MCG(S_{g,b})$ contains not only a lot of "weird" elements, but also is equipped with nice structures. By that I mean that $MCG(S_{g,b})$ contains a lot of abelian subgroups.
Finding some of these abelian subgroups is relatively simple, for example Dehn-Twists along non-intersecting curves generate abelian subgroups isomorphic to $\mathbb{Z}^n$. But is there a way to find maximal rank abelian subgroups?
I know that there are bounds on the rank of abelian subgroups in $MCG(S_{g,b})$ by results of Atalan, but are there any statements on existence of a maximum rank abelian subgroup or how one would find them?
For example, it is also "well-known" that $MCG(S_{g,0})$ can be generated by 2g+1 Dehn-Twists, although in a non-abelian way. So what in that case, how would one find a maximum set of such Dehn-Twists such that their generated subgroup would be abelian? And in that case, is it really a maximum rank abelian subgroup of $MCG(S_{g,0})$?
Best Answer
The set of all maximal rank free abelian subgroups is well known as an application of Ivanov's subgroup theorem, and is described as follows.
First, cut $S_{g,0}$ along simple closed curves into pairs of pants, one-holed tori, four holed spheres, and annuli, such that each simple closed curve $\gamma$ in the collection bounds an annulus on one side of $\gamma$ and one of the other three types on the other side of $\gamma$. The number of non-pants components of this decomposition will be $3g-3$; list them as $C_1,...,C_{3g-3}$.
For each $i=1,...,3g-3$ define a homeomorphism $h_i$ which is the identity outside of $C_i$, and when restricted to $C_i$ is: a Dehn twist power if $C_i$ is an annulus; or a pseudo-Anosov homeomorphism (up to isotopy) if $C_i$ is a one-holed torus or four holed sphere. The mapping classes of these homeomorphisms freely generate a maximal rank free abelian group, and all maximal rank free abelian groups arise in this manner.
A similar generalization holds for $S_{g,b}$.