Quantifying algebraic properties of surface groups and 3-manifold groups

Abstract

OF THE DISSERTATION Quantifying Algebraic Properties of Surface Groups and 3-Manifold Groups by Priyam Patel Dissertation Director: Feng Luo A group G is residually finite (RF) if for every nontrivial element g ∈ G, there exists a finite index subgroup G′ of G such that g / ∈ G′. A group G is called locally extended residually finite (LERF) if for any finitely generated subgroup S of G and any g ∈ 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.