Let K=Q(theta) be the algebraic number field formed by adjoining theta to the rationals where theta is a real root of an irreducible monic cubic polynomial f(x) in Z[x]. If theta is not the cube root of a rational integer, we call the field K a non-pure cubic field, and if K doesn't contain the conjugates of theta, we call K a non-normal cubic field. A method described by Martinet and Payan allows us to construct such fields from elements of a quadratic field. In this work, we examine such non-normal, non-pure cubic fields and their normal closures, using algorithms in Mathematica to compute various invariants of these fields. In addition, we prove general results relating the ranks of the ideal class groups of the rings of integers of these cubic fields to those of their normal closures. / Ph. D.

Thesis advisor: Martin J. Bridgeman / Thesis advisor: Ian Biringer / The first part of this dissertation is on the quasiconformal homogeneity of surfaces. In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces. We then move on to identities on hyperbolic manifolds. We study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSU(1,1). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli space (i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to this question is to use sewing techniques which requires that one considers the action on the level of homomorphisms, and for surfaces with boundary. In this paper we consider the case of the one-holed torus with boundary condition, and we determine regions where the action is ergodic. This uses a combination of techniques developed by Goldman, and Pickrell and Xia. The basic result is an analogue of the result of Goldman's at the level of moduli.

In this thesis we deal with combinatorial and geometric properties of arc complexes and triangulation graphs, and we will provide some applications to the study of the mapping class group and to the Teichmüller theory of a bordered surface. The thesis is divided into two parts. In the former we deal with the problem of combinatorial rigidity of arc complexes. In the latter we study some large-scale properties of the arc complex and the 1-skeleton of its dual, the so-called ideal triangulation graph.

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If we adjoin the cube root of a cube free rational integer <i>m</i> to the rational numbers we construct a cubic field. If we adjoin the cube roots of distinct cube free rational integers <i>m</i> and <i>n</i> to the rational numbers we construct a bicubic field. The number theoretic invariants for the cubic fields and their normal closures are well known. Some work has been done on the units, classnumbers and other invariants of the bicubic fields and their normal closures by Parry but no method is available for calculating those invariants. This dissertation provides an algorithm for calculating the number theoretic invariants of the bicubic fields and their normal closure. Among these invariants are the discriminant, an integral basis, a set of fundamental units, the class number and the rank of the 3-class group. / Ph. D.

Soit E une courbe elliptique sur C ayant multiplication complexe (CM) par l’ordre maximal OK d’un corps quadratique imaginaire K. Le premier théorème principal de la multiplication complexe affirme que le corps K(j(E)), obtenu en adjoignant à K le j-invariant de E, est égal au corps de classes de Hilbert de K, confer Cox [11, Theorem 11.1]. Notons que lorsque E est définie sur Q, le corps de classes de Hilbert K(j(E)) est égal à K et le groupe des classes ClK est trivial. Se pose alors le problème de déterminer les corps quadratiques totalement imaginaires K pour lesquels la courbe elliptique à multiplication complexe par OK correspondante est définie sur Q. De façon équivalente, il s’agit de trouver tous les corps quadratiques imaginaires dont le groupe des classes est trivial. Ce problème est connu sous le nom de problème du nombre de classes 1 de Gauss et a été résolu par Heegner en 1952 [16], Baker en 1967 [2] et Stark en 1967 [41]; les corps quadratiques imaginaires dont le groupe des classes est trivial sont les corps Q(racine carrée−d), où d e {3, 4, 7, 8, 11, 19, 43, 67, 163}. Dans les années ’50, Shimura et Taniyama [39] ont généralisé le premier théorème principal de la multiplication complexe aux variétés abéliennes. On dit qu’une variété abélienne A de dimension g a multiplication complexe si son anneau d’endomorphismes contient un ordre d’un corps CM de degré 2g. Soit K un corps CM de degré 2g et d’ordre maximal OK et soit un type CM de K. Soit A une variété abélienne complexe simplement polarisée de dimension g ayant multiplication complexe par OK. Le premier théorème principal de la multiplication complexe dans ce cadre affirme que le corps de classes H du corps du modules M de la variété abélienne simplement polarisée A est une extension non ramifiée du corps reflex Kr de K. De plus, le corps des classes H correspond au groupe d’idéaux I0(.r) (voir page 17) qui ne dépend que de (K,.), confer Théorème 1.5.6. Notons que le premier théorème de la multiplication complexe implique que si la variété abélienne polarisée A est définie sur Kr, le groupe des classes CM IKr/I0(.r) est trivial. Comme dans le cas des courbes elliptiques, on peut alors chercher à déterminer les couples CM (K,.) pour lesquels les variétés abéliennes correspondantes sont définies sur Kr. De fa¸con équivalente, il s’agit de déterminer les couples CM (K,.) dont le groupe des classes CM, IKr/I0(.r), est trivial. Dans cette thèse, on résout ce problème dans le cas des corps CM quartiques imaginaires (voir Chapitre 2) ainsi que dans celui des corps CM sextiques contenant un corps quadratique imaginaire (voir Chapitre 3). Enfin, on peut se demander quels sont les corps CM pour lesquels la variété abélienne simple à multiplication complexe admet Q comme corps de module. Murabayashi et Umegaki [31] ont déterminé les corps quartiques CM correspondant aux surfaces abéliennes simples à multiplication complexe de corps du module Q. Dans le chapitre 4, on détermine les corps CM sextiques correspondant aux variétés abéliennes simples à multiplication complexe de dimension 3 de corps du module Q. / Let E be an elliptic curve over C with complex multiplication (CM) by the maximal order OK of an imaginary quadratic field K. The first main theorem of complex multiplication for elliptic curves then states that the field extension K(j(E)), obtained by adjoining the j-invariant of E to K, is equal to the Hilbert class field of K, see Theorem 11.1 in Cox [11]. Note that if E is defined over Q, then the Hilbert class field K(j(E)) is equal to K, which implies that the class group ClK is trivial. We can ask for which imaginary quadratic fields K the corresponding elliptic curve with CM by OK is defined over Q. This is equivalent to asking to find all imaginary quadratic fields with trivial class group ClK. This problem is known as Gauss’ class number one problem, which was solved by Heegner in 1952 [16], Baker in 1967 [2], and Stark in 1967 [41]. The imaginary quadratic fields with trivial class group are the fields Q(V−d) with d E {3, 4, 7, 8, 11, 19, 43, 67, 163}. In the 1950’s, Shimura and Taniyama [39] generalized the first main theorem of CM for elliptic curves to abelian varieties. We say that an abelian variety A of dimension g has CM if the endomorphism ring of A contains an order of a CM field of degree 2g. Let K be a CM field of degree 2g with maximal order OK, and let K be a CM type of K. Let A be a polarized simple abelian variety over C of dimension g that has CM by OK. Then the first main theorem of CM says that the field of moduli M of the polarized simple abelian variety A gives an unramified class field H over the reflex field Kr of K. Moreover, the class field H corresponds to the ideal group I0(?r) (see page 17), which only depends on (K,?), see Theorem 1.5.6. Note that the first main theorem of CM implies that if the polarized abelian variety A is defined over Kr, then the CM class group IKr/I0(?r) is trivial. As in the elliptic curve case, we can ask for which CM pairs (K,?) the corresponding CM abelian varieties are defined over Kr. Equivalently, we can ask for which CM pairs (K,?) the CM class group IKr/I0(?r) is trivial. In this thesis we give an answer to this problem for quartic CM fields (see Chapter 2), and for sextic CM fields containing an imaginary quadratic field (see Chapter 3). Furthermore, we can ask for which CM fields the corresponding simple CM abelian varieties have field of moduli Q. Murabayashi and Umegak [31] determined the quartic CM fields that correspond to a simple CM abelian surface with field of moduli Q. In Chapter 4, we determine the sextic CM fields that correspond to a simple CM abelian threefold with field of moduli Q.

In this thesis we build on the theory concerning the metric geometry of relatively hyperbolic and mapping class groups, especially with respect to the difficulty of embedding such groups into Banach spaces. In Chapter 3 (joint with Alessandro Sisto) we construct simple embeddings of closed graph manifold groups into a product of three metric trees, answering positively a conjecture of Smirnov concerning the Assouad-Nagata dimension of such spaces. Consequently, we obtain optimal embeddings of such spaces into certain Banach spaces. The ideas here have been extended to other closed three-manifolds and to higher dimensional analogues of graph manifolds. In Chapter 4 we give an explicit method of embedding relatively hyperbolic groups into certain Banach spaces, which yields optimal bounds on the compression exponent of such groups relative to their peripheral subgroups. From this we deduce that the fundamental group of every closed three-manifold has Hilbert compression exponent one. In Chapter 5 we prove that relatively hyperbolic spaces with a tree-graded quasi-isometry representative can be characterised by a relative version of Manning's bottleneck property. This applies to the Bestvina-Bromberg-Fujiwara quasi-trees of spaces, yielding an embedding of each mapping class group of a closed surface into a finite product of simplicial trees. From this we obtain explicit embeddings of mapping class groups into certain Banach spaces and deduce that these groups have finite Assouad-Nagata dimension. It also applies to relatively hyperbolic groups, proving that such groups have finite Assouad-Nagata dimension if and only if each peripheral subgroup does.

Nous montrons que les morphismes du groupe de tresses à n brins dans le mapping class group d'une surface de bord éventuellement non vide et de genre inférieur ou égal à n/2 sont soit cycliques (i.e. dont l'image est un groupe cyclique), soit des transvections de monodromie géométriques (i.e. à multiplication près par un élément du centralisateur de l'image, un générateur standard du groupe de tresses est envoyé sur un twist de Dehn, et deux générateurs standards consécutifs sont envoyés sur deux twists de Dehn le long de deux courbes s'intersectant en un point). En corollaire, nous déterminons les endomorphismes, les endomorphismes injectifs, les automorphismes et le groupe d'automorphisme des groupes suivants : le groupe de tresses à n brins lorsque n est supérieur ou égal à 6, le mapping class group de toute surface de genre supérieur ou égal à 2. Pour chacun des énoncés impliquant le mapping class group, nous étudions deux cas : lorsque le bord est fixé point par point ou seulement composante par composante. Nous décrivons également l'ensemble des morphismes entre différents groupes de tresses dont le nombre de brins diffèrent d'au plus un, et l'ensemble des morphismes entre mapping class groups de surfaces (de bord éventuellement non vide) dont les genres (supérieurs ou égal à 2) différent d'au plus un.

The elasticity of a Krull domain R is equivalent to the elasticity of the block monoid B(G,S), where G is the divisor class group of R and S is the set of elements of G containing a height-one prime ideal of R. Therefore the elasticity of R can by studied using the divisor class group. In this dissertation, we will study infinite divisor class groups to determine the elasticity of the associated Krull domain. The results will focus on the divisor class groups Z, Z(p infinity), Q, and general infinite groups. For the groups Z and Z(p infinity), it has been determined which distributions of the height-one prime ideals will make R a half-factorial domain (HFD). For the group Q, certain distributions of height-one prime ideals are proven to make R an HFD. Finally, the last chapter studies general infinite groups and groups involving direct sums with Z. If certain conditions are met, then the elasticity of these divisor class groups is the same as the elasticity of simpler divisor class groups.