I want to find a subgroup $H$ of the (orientation preserving) mapping class group $G=MCG(g,n)$ of a surface with genus $g$ and $n$ boundary components that satisfies the following properties:
- $[G:H]<\infty$
- $H$ is solvable
- Every subgroup of $H$ is finitely generated
I've had some ideas but none of them satisfy all three (that I know of). For instance, I've thought about taking a subgroup generated by Dehn twists around disjoint curves. This will obviously satisfy 2 and 3 but I believe it fails 1. The only other subgroup that springs to mind is the torelli subgroup, but I have no idea if it satisfies any of these properties let alone all 3.
So, is there a subgroup satisfying these 3 properties? Thanks for any help
Best Answer
Outside of low genus cases this does not happen.
$MCG(0,0),MCG(0,1),MCG(0,2),MCG(0,3)$ are abelian, and hence already solvable and also satisfy "3"(since the groups are finitely generated).
All other mapping class groups do not have finite index solvable subgroups. These mapping class groups have nonabelian free subgroups, and so any finite index subgroup will contain also contain a nonabelian free subgroup. There are many ways to see that they contain nonabelian free subgroups. One way is to calculate the mapping class group of torus(with puncture or boundary component) and 4-holed sphere(see the A primer on mapping class groups by Farb and Margalit). These are virtually nonabelian free and appear as essential subsurfaces of all higher complexity surfaces, so the mapping class groups of the higher complexity surfaces contain these natural mapping class groups as subgroups.
Further every solvable subgroup of mapping class groups of surfaces is actually virtually abelian and these groups basically only appear in the "obvious" way: groups generated by mapping classes with disjoint support on the surface. This can be found in Abelian and solvable subgroups of the mapping class group by Birman, Lubotzky, and McCarthy.