The normal closure of a big Dehn twist in the mapping class group
Joint with Vincent Guirardel and Denis Osin.
We are interested in quotients of the mapping class group. Influenced by the theory of lattices in Lie groups, Farb and Ivanov asked what is the normal closure of a pseudo-Anosov element. The idea is that pseudo-Anosov elements reflect the hyperbolic character of MCG, whereas reducible elements reflect the higher rank behaviour.
Theorem 1 (Dahmani-Guirardel-Osin) There exists such that the normal closure of the -th power a pseudo-Anosov element is free and purely pseudo-Anosov.
This is ping-pong.
Fix a simple closed curve on a closed surface . Let be the corresponding Dehn twist. What happens with the nroaml closure of a high power ?
Koberda: given a finite family of distinct simple closed curves, the group generated by is a right-angled Artin group in its natural presentation (Dehn twists around disjoint curves commute) for large enough.
Is a right-angled Artin group ? No. Artin groups have too many automorphisms. For instance, one which maps a genrator to its inverse, without changing other generators. This cannot arise from a surface homeomorphism.
Brendle-Margalit: is the extended MCG acting by conjugation.
Theorem 2 There exists such that, for , has a partially commutative presentation over some conjugates of . I.e., there is a suitable infinite set of generators which are conjugates of , and relators of the form , where are words in .
In fact, a relator arises iff the corresponding curves are disjoint.
3.1. The pseudo-Anosov case
We use rotation families of subgroups. Let be a group acting on a hyperbolic space . Let be loxodromic in , with axis . Assume hat the collection of conjugates of has the following property: axes are
- either equal,
- or do not overlap on a length .
Let us glue a hyperbolic cone of large radius, with infinite angle at the tip, to each axis. Do this equivariantly. In the resulting space, loxodromic translations become rotations (whence the term rotation family). Then, for large enough, the group is free.
Indeed, let a ball centered at one cone apex grow, adding progressively its images under rotations around the other apices it encounters. The picture obtained is a tree, since branches are local quasi-geodesics.
This argument applies to the MCG action on the curve complex
3.2. The Dehn twist case
Bestvina-Bromberg-Fujiwara: there exists a normal finite index subgroup in MCG and a -invariant finite coloring of the set of simple close curves of such that two curves with the same color intersect.
Theorem 3 (Bestvina-Bromberg-Fujiwara) There exists a finite collection of quasi-trees on which acts and for which every Dehn twist is loxodromic on one tree and elliptic on all others. Furthermore, axes satifies the rotation family condition.
If is loxodromic on the first quasi-tree , we get that is free. We want the full normal subgroup, so we need to perform the rotation family argument simultaneously on all quasi-trees. There is a difficulty: one needs a control on on the secondary quasi-trees, a bound on its behaviour on cones. This uses more “projection” theory.