A new filtration of the Magnus kernel
| dc.contributor.advisor | Harvey, Shelly | |
| dc.contributor.committeeMember | Cochran, Tim D. | |
| dc.contributor.committeeMember | Riviere, Beatrice M. | |
| dc.creator | McNeill, Reagin | |
| dc.date.accessioned | 2013-09-16T15:58:08Z | |
| dc.date.accessioned | 2013-09-16T15:58:10Z | |
| dc.date.available | 2013-09-16T15:58:08Z | |
| dc.date.available | 2013-09-16T15:58:10Z | |
| dc.date.created | 2013-05 | |
| dc.date.issued | 2013-09-16 | |
| dc.date.submitted | May 2013 | |
| dc.date.updated | 2013-09-16T15:58:10Z | |
| dc.description.abstract | For a oriented genus g surface with one boundary component, S_g, the Torelli group is the group of orientation preserving homeomorphisms of S_g that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F'' where F=π_1(S_g) and F'' is the second term of the derived series. I show that the kernel of the Magnus representation, Mag(S_g), is highly non-trivial and has a rich structure as a group. Specifically, I define an infinite filtration of Mag(S_g) by subgroups, called the higher order Magnus subgroups, M_k(S_g). I develop methods for generating nontrivial mapping classes in M_k(S_g) for all k and g≥2. I show that for each k the quotient M_k(S_g)/M_{k+1}(S_g) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally I show that for g≥3 the quotient M_k(S_g)/M_{k+1}(S_g) surjects onto an infinite rank torsion free abelian group. To do this, I define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.citation | McNeill, Reagin. "A new filtration of the Magnus kernel." (2013) Diss., Rice University. <a href="https://hdl.handle.net/1911/72006">https://hdl.handle.net/1911/72006</a>. | |
| dc.identifier.slug | 123456789/ETD-2013-05-458 | |
| dc.identifier.uri | https://hdl.handle.net/1911/72006 | |
| dc.language.iso | eng | |
| dc.rights | Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder. | |
| dc.subject | Lower central series | |
| dc.subject | Surfaces | |
| dc.subject | Mapping class groups | |
| dc.subject | Pure braids | |
| dc.subject | Magnus kernel | |
| dc.subject | Free groups | |
| dc.subject | Johnson subgroups | |
| dc.title | A new filtration of the Magnus kernel | |
| dc.type | Thesis | |
| dc.type.material | Text | |
| thesis.degree.department | Mathematics | |
| thesis.degree.discipline | Natural Sciences | |
| thesis.degree.grantor | Rice University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy |