Most interesting 3d Topological Quantum Field Theories (TQFTs) are constructed by starting with algebraic data, usually in the form of some kind of category. This category typically comes from an area of mathematics different from 3-manifold topology, and its topological nature can be hard to understand. This dissertation reverses the process, at least in one simple example, by constructing a Spin TQFT from pure topology and then uncovering some interesting categories.

The topology used to construct the Spin TQFT is entirely classical. If $(M, s)$ is a closed Spin 3-manifold, then an embedded surface $\Sigma\subset M$ inherits a $\Pin^-$ structure, $s|_{\Sigma}$, from $(M, s)$. If $\Sigma$ and $\Sigma'$ represent the same class in $H_2(M;\bbZ/2)$, then $s|_{\Sigma}$ and $s|_{\Sigma'}$ are isomorphic. If $t$ is a $\Pin^-$ structure on $\Sigma$, there is a classical invariant of the isomorphism type of $(\Sigma, t)$, denoted $\beta(\Sigma, t)$, that is an eight root of unity. One can therefore form the Spin 3-manifold invariant

\[

Z_3(M, s) := \frac{1}{2^{b_0(M)}} \sum_{[\Sigma]\in H_2(M;\bbZ/2)} \beta(\Sigma, s|_{\Sigma}).

\]

It turns out that this invariant fits into a Spin TQFT. The detailed construction of this TQFT is the subject of this dissertation.

By a theorem of Kirby and Melvin, $Z_3$ is very much related to the Ising categories. In extending the TQFT for $Z_3$, one encounters a category (associated with the bounding Spin circle) which has most of the same properties as the Ising categories. One also encounters a category (associated with the interval) from which $Z_3$ can be reconstructed in the style of Turaev-Viro and Barrett-Westbury. Because $Z_3$ is a Spin TQFT, these categories are linear over super vector spaces. In fact, they are realized as the module categories of certain explicit super algebras.

In this thesis, we explore many aspects of groups acting on trees and on products of trees. These ideas are central to the field of geometric group theory, the study of infinite groups by their large-scale behavior. Many of our techniques are algebraic and arithmetic in nature. Most of this work is motivated by the following question:

\begin{ques} If $G$ is the fundamental group of a closed surface, can $G$ act freely on a locally compact product of trees? \end{ques}

In fact, the more general question of whether a hyperbolic group which is not virtually free can act properly on a locally compact Euclidean building is open. Of course such an action gives a cubulation of the group in perhaps the simplest combinatorial type of cube complex, so this is a necessary condition on the group. For instance, which right-angled Coxeter or Artin groups admit proper actions on products of trees?

In a landmark paper of 1987, Gromov defines the class of hyperbolic groups, and nearly every branch of modern geometric group theory is represented in Gromov's original paper. However, Dehn's work in 1910 and 1911 may be viewed as the earliest work in geometric group theory. For several decades, Dehn's work was viewed with a more combinatorial lens, until Gromov emphasized the intimate connection with hyperbolic geometry.

Thurston's work in 3-dimensional topology together with Gromov's hyperbolic groups aligned topology and group theory. One might ask \emph{to what extent is a 3-manifold determined by its fundamental group?} The Poincar e conjecture is a precise version of this question, famously resolved by Perelman. In a sense, the fundamental group of a 3-manifold is an excellent replacement for the manifold itself, and in the finite-volume hyperbolic case, the fundamental group miraculously contains all of the metric information!

For closed hyperbolic surfaces, the fundamental group knows only the genus, and there is a high-dimensional space of representations a given surface can support. For a 4-manifold, the fundamental group says relatively little about the topology or the geometry. Every countable group can occur as the fundamental group of a 4-manifold, and there is a vast land of simply connected four-manifolds. However, it is interesting to note that, at least up to finite index, closed hyperbolic 3-manifold fundamental groups are determined by the data of an automorphism of a surface group. Upon replacing ``hyperbolic 3-manifolds'' with ``discrete subgroups of $\PSL_2(\C)$'', it is tempting to consider ``subgroups of $\PSL_2(\C)$''. This is the approach we take in this thesis. We observe that $\PSL_2(\overline{\Q})$ acts properly discontinuously and cocompactly on a ``restricted'' product of infinitely many hyperbolic planes, 3-spaces, and finite valence trees. By allowing transcendental entries, we get a similar statement, but not all of the trees will have finite-valence. But in fact, subgroups of $\PSL_2(\C)$ with cofinite volume are actually already conjugate into $\PSL_2(\overline{\Q})$ anyway. That is to say, every hyperbolic 3-manifold of finite volume comes equipped with a canonical action on a product of infinitely many finite-valence trees.

One can ask which properties of discrete subgroups of $\PSL_2(\C)$ might carry over to these more general subgroups of $\PSL_2(\C)$. Some celebrated properties of Kleinian groups include coherence, tameness, LERFness. It is unknown to what extent each of these properties might hold in this more general setting.

The main result of this dissertation shows that every orientable closed 3-manifold admits a nonzero degree map onto at most finitely many homeomorphically distinct non-geometric prime 3-manifolds. Furthermore, for any integer d > 0, every orientable closed 3-manifold admits a map of degree d onto only finitely many homeomorphically distinct 3-manifolds. This answers a question of Yongwu Rong. The finiteness of JSJ piece of the targets under nonzero degree maps was known earlier by the results of Soma and Boileau–Rubinstein–Wang, and a new proof is provided is this dissertation. We also prove analogous results for dominations rela-tive to boundary. As an application, we describe the degree set of dominations onto integral homology 3-spheres.

We construct an invariant called guts for second homology classes in irreducible 3-manifolds with toral boundary and non-degenerate Thurston norm. We prove that guts of second homology classes in each Thurston cone are invariant under a natural condition. We show that the guts of different homology classes are related by sutured decompositions. As an application, an invariant of knot complements is given and is computed in a few interesting cases.

The minimal volume of orientable hyperbolic manifold with a given number of cusps has been found for 0,1,2,4 cusps, while the minimal volume of 3-cusped orientable hyperbolic manifolds remains unknown. By using guts in sutured manifolds and pared manifolds, we are able to show that for an orientable hyperbolic 3-manifold with 3 cusps such that every second homology class is libroid, its volume is at least 5.49... = 6 times Catalan’s constant.

In the final chapter, we develop a method for controlling the upper bound of the Euler characteristic of surfaces in sufficiently long Dehn fillings. By using this method, we show that for a compact orientable irreducible acoannular 3-manifold with toral boundary, properly norm-minimizing surfaces capped off with disks in sufficiently long fillings are still properly norm-minimizing. We also show that if a sufficiently long Dehn surgery on a link in a product sutured manifold yields the same product sutured manifold, then this link is horizontal in the product sutured manifold. Combining these results, we prove that for a compact orientable irreducible acoannular 3-manifold with toral boundary, the cores of its sufficiently long fibered Dehn fillings are either horizontal or vertical. Moreover, we provide a new way to prove that there are finitely many links with a given complement such that no component is unknotted and no 2-component sublink is coaxial. Another result coming from this method is that all sufficiently long fillings of an orientable, irreducible, boundary-irreducible-irreducible, acoannular 3-manifold with toral boundary are still irreducible, boundary-irreducible-irreducible, acoannular.

In addition to classical knot theory, low-dimensional topology is also concerned with a variety of alternative classes of knot-like objects, including knotoids, braids, and string links. In this thesis we explore several of these theories as well as certain connections between them. While our results cover several different types of structures, they nonetheless build on each other.

The theory of knotoids is an extension of classical knot theory whereby knot diagrams are allowed to have two endpoints instead of being a closed loop. One of the most important knotoid invariants is the height h, which measures how far a knotoid is from being a classical knot. After defining the signed versions h_+ and h_- of the height, we prove that together they determine the unsigned height by the simple formula h_+ + h_- = h, and we demonstrate a few applications.

String links, which are tangles that have been normalized with respect to the locations of the endpoints of the strands, come with a natural monoid operation that generalizes both connected summation of knots and composition of braids. Previous work on strings links has shown that the string link monoids are cancellative up to multiplication with units. Here we strengthen that result by proving the freeness of unit multiplication on either side, thus answering a question of Blair–Burke–Koytcheff and implying full cancellativity of the string link monoids. In the same section, we prove a weaker version of the well-known cosmetic crossing conjecture and explore applications of knotoids to braid groups.

We also study an infinite version of string links, which we call Z-tangles. These Z-tangles are interesting in part because a version of Birman–Hilden theory applies to Z-tangles as they relate to knotoids. Specifically, there is a natural forgetful function from the set of knotoids to the set of Z-tangles up to isotopy. We give several pieces of evidence for the conjectural injectivity of this Birman–Hilden map, most notably by using some of the results of earlier chapters to identify classes of knotoids that will each be uniquely determined by their image.

This dissertation concerns two subjects that the author explored in his doctoral research: the Burau representation of the braid groups and moduli spaces of Euclidean cone surfaces. We present some general theory on each subject and the connection that eventually ties the two together. In particular, we see original results of the author that leverage the geometry of the moduli spaces of flat cone spheres to identify the kernel of certain evaluations of the Burau representation at roots of unity. Figures abound.