Parages, Leire (2026) From Curves to Symmetries in Mapping Class Groups and Handlebodies. Bachelor's Thesis, Mathematics.
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Abstract
Simple closed curves are one of the simplest objects one can draw on a surface, yet they encode most of the algebraic structure of the mapping class group Mod(Σg). This thesis studies that connection from two directions. First, we show how Mod(Σg) is built out of Dehn twists: we prove the Dehn–Lickorish theorem, which says that finitely many Dehn twists generate the whole group, and we describe the local and global relations, such as the braid, commutation, and lantern relations, that hold between them. We then use the Alexander method, which recovers a mapping class from its action on a filling collection of curves, as a tool for verifying such relations. Second, we turn to the handlebody subgroup Hg ⊂ Mod(Σg), consisting of the mapping classes that extend over a genus-g handlebody Vg. We show that Hg is exactly the stabiliser of the set of meridian curves, and we prove that Hg is self-normalising in Mod(Σg). Building on this, we study the intersection Hg ∩ ϕHgϕ−1 of two handlebody subgroups glued by a mapping class ϕ, and show how its size, infinite, finite, or trivial, is governed by the distance d(ϕ) between the two meridian sets in the curve complex C(Σg). An explicit example, built from a pseudo-Anosov product of Dehn twists, illustrates each of these regimes concretely.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Karakurt, C. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 28 Jul 2026 06:59 |
| Last Modified: | 28 Jul 2026 06:59 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/37471 |
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