| This dissertation consists of two chapters, one on 3-manifold topology and the other on geometric group theory. These chapters are somewhat independent of each other.;In the first chapter, we investigate how a connected compact 3-manifold Y can be embedded into an orientable closed 3-manifold M . We show that the following two conditions are equivalent: (i) Y can be embedded in M so that the closure of the complement of the image of Y is a union of handlebodies; and (ii) Y can be embedded in M so that every embedded closed loop in M can be isotoped to lie within the image of Y. Our result can be regarded as a common generalization of Fox's re-embedding theorem and Bing's characterization of the 3-sphere, as well as more recent results of Hass and Thompson, and Kobayashi and Nishi.;In the second chapter, we investigate the girth of finitely generated groups and its properties. The girth of a finitely generated group Gamma is the supremum of the girth of Cayley graphs for Gamma over all finite generating sets. We first establish a dynamical criterion for a group Gamma to have infinite girth. We then apply this criterion to certain classes of groups, such as subgroups of mapping class groups and discrete convergence groups, and show that a group from these classes is either a virtually abelian group or a non-cyclic group with infinite girth. This result shows that the dichotomy between groups with finite girth and infinite girth coincides with the dichotomy that appears in the Tits-alternative for these groups. |
