A group G is residually finite (RF) if for every nontrivial element g in G, there exists a finite index subgroup G' of G such that g is not in G'. A group G is called locally extended residually finite (LERF) if for any finitely generated subgroup S of G and any g in G -S, there exists a finite index subgroup G' of G which contains S but not g. Quantifying these algebraic finiteness properties refers to bounding the indexes of the finite index subgroups G' in each of the definitions above. In this dissertation we quantify Peter Scott's theorem that surface groups are LERF in terms of geometric data. In the process, we will quantify the fact that surface groups are residually finite and quantify another result by Scott that any closed geodesic in a surface lifts to an embedded loop in a finite cover. We also extend the methods used in the 2-dimensional case to quantify the residual finiteness of particular 3-manifold groups.
Subject (authority = RUETD)
Topic
Mathematics
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.