Maccoll, JI;
(2021)
w-Cycles in Surface Groups.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
For w an element in the fundamental group of a closed, orientable, hyperbolic surface Ω which is not a proper power, and Σ a surface immersing in Ω, we show that the number of distinct lifts of w to Σ is bounded above by -χ(Σ). In special cases which can be characterised by interdependencies of the lifts of w, we find a stronger bound, whereby the total degree of covering from curves in Σ representing the lifts to the curve representing w is also bounded above by -χ(Σ). This is achieved by a method we introduce for decomposing surfaces into pieces that behave similarly to graphs, and using them to estimate Euler characteristics using a stacking argument of the kind introduced by Louder and Wilton. We then consider some consequences of these bounds for quotients of orientable surface groups by a single element. We demonstrate ways in which these groups behave analogously to one-relator groups; in particular, the ones with torsion are coherent (i.e. all finitely-generated subgroups have finite presentations), and those without torsion possess the related property of non-positive immersions as introduced by Wise.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | w-Cycles in Surface Groups |
| Event: | UCL (University College London) |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10139119 |
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