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vii, 124 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation is a study of the relationship between minimal dynamical systems on the product of the Cantor set ( X ) and torus ([Special characters omitted]) and their corresponding crossed product C *-algebras. For the case when the cocyles are rotations, we studied the structure of the crossed product C *-algebra A by looking at a large subalgebra A x . It is proved that, as long as the cocyles are rotations, the tracial rank of the crossed product C *-algebra is always no more than one, which then indicates that it falls into the category of classifiable C *-algebras. In order to determine whether the corresponding crossed product C *-algebras of two such minimal dynamical systems are isomorphic or not, we just need to look at the Elliott invariants of these C *-algebras. If a certain rigidity condition is satisfied, it is shown that the crossed product C *-algebra has tracial rank zero. Under this assumption, it is proved that for two such dynamical systems, if A and B are the corresponding crossed product C *-algebras, and we have an isomorphism between K i ( A ) and K i ( B ) which maps K i (C(X ×[Special characters omitted])) to K i (C( X ×[Special characters omitted])), then these two dynamical systems are approximately K -conjugate. The proof also indicates that C *-strongly flip conjugacy implies approximate K -conjugacy in this case. We also studied the case when the cocyles are Furstenberg transformations, and some results on weakly approximate conjugacy and the K -theory of corresponding crossed product C *-algebras are obtained. / Committee in charge: Huaxin Lin, Chairperson, Mathematics Daniel Dugger, Member, Mathematics; Christopher Phillips, Member, Mathematics; Arkady Vaintrob, Member, Mathematics; Li-Shan Chou, Outside Member, Human Physiology

There has often been a connection between music and mathematics. The world of musical composition is full of combinations of orderings of different musical notes, each of which has different sound quality, length, and em phasis. One of the more intricate composition styles is twelve-tone music, where twelve unique notes (up to octave isomorphism) must be used before they can be repeated. In this thesis, we aim to show multiple ways in which mathematics can be used directly to compose twelve-tone musical scores.

The nature of this paper is expository. The purpose is to present the fundamental material concerning actions of infinite discrete groups on the n-sphere and pseudo-Riemannian space forms based on the works of Gehring, Martin and Kulkarni and provide appropriate examples. Actions on the n-sphere split it into ordinary and limit sets. Assuming, additionally, that a group acting on the n-sphere has a certain convergence property, this thesis includes conditions for the existence of a homeomorphism between the limit set and the set of Freudenthal ends, as well as topological and quasiconformal conjugacy between convergence and Mobius groups. Since the certain pseudo-Riemannian space forms are diffeomorphic to non-compact spaces, the work of Hambleton and Pedersen gives conditions for the extension of discrete co-compact group actions on pseudo-Riemannian space forms to actions on the sphere. An example of such an extension is described. / Thesis / Master of Science (MSc)

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The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L_p compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.

Dans la thèse présente, nous nous intéressons à l'étude de la relative hyperbolicité des produits semi-direct des produits libres, ainsi que le problème de conjugaison pour certains automorphismes de ces produits libres.Plus précisement, pour un produit libre $$G=G_1astdotsast G_past F_k$$ un automorphisme $phi$ est intitulé atoroidal s'il ne fixe pas (ni aucune de ses puissances) la classe de conjugaison d'un élément hyperbolique de $G$. Cet automorphisme est appelé completement irréductible si le système de facteurs libres est le plus grand qui est fixé par toutes les puissances de cet automorphisme. Il est appelé toral si pour tous les $i$, il existe $g_iin G$ tel que ${rm ad}_{g_i}circ phi|_{G_i}$ est identité sur le facteur libre $G_i$. Nous disons qu'il a la condition centrale si pour chaque $i$, il existe $g_iin G$ conjugue $phi(G_i)$ à $G_i$, et s'il existe un élément non trivial de $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$ qui est central dans $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$.Nous prouvons, dans le Théorème 4.28, que si $phi$ est atoroidal et completement irréductible, et si le produit libre est non-elementaire ($kgeq 2$ ou $ p+k geq 3$), le groupe $Grtimes_phi mathbb{Z}$ est relativement hyperbolique (relativement a des suspensions de chaque $G_i$). Après, dans le Théorème 6.10, nous prouvons le même résultat si $phi$ est atoroidal avec la condition centrale. Nous prouvons aussi dans le Théorème 7.21 que si tous les $G_i$ sont abelien, le problème de conjugaison est solvable pour les automorphismes atoroidaux, toraux. Ces sont des analogues du résultat de Brinkmann [7] (celui qui a donné le résultat d'hyperbolicité pour les groupes libres), et du résultat de Dahmani [12] (celui qui a résolu le problème de conjugaison des automorphismes hyperboliques). / In this thesis, we are interested in the study of the relative hyperbolicity of the suspensions of free products, as well as the conjugacy problem of certain automorphisms of free products.To be more precise, given a free product $$G=G_1astdotsast G_past F_k$$ an automorphism $phi$ is said atoroidal if no power fixes the conjugacy class of an hyperbolic element. It is called fully irreducible if the given free factor system $[G_1],dots,[G_p]$ is the largest one that is fixed by every power of the automorphism. It is said toral if for all $i$, there exists $g_iin G$ such that ${rm ad}_{g_i}circ phi|_{G_i}$ is the identity on the free factor $G_i$. It is said to have central condition if for each $i$, there exists $g_iin G$ conjugating $phi(G_i)$ to $G_i$, and if there exists a non-trivial element of $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$ that is central in $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$.We prove, in Theorem 4.28, that if $phi$ is atoroidal and fully irreducible, and if the free product is non-elementary ($kgeq 2$ or $ p+k geq 3$), the group $Grtimes_phi mathbb{Z}$ is relatively hyperbolic (relative to the mapping torus of each $G_i$). Then in Theorem 6.10 we prove the same result holds if $phi$ is atoroidal with central condition. We also prove in Theorem 7.21 that if all $G_i$ are abelian, the conjugacy problem is solvable for toral atoroidal automorphisms. These are analogue of the result of Brinkmann [7] (which gave the hyperbolicity result for free groups) and the result of Dahmani [12] (which solved the conjugacy problem of hyperbolic automorphisms).

Nós estudaremos o comportamento de difeomorfismos do círculo, tanto do ponto de vista combinatório quanto do ponto de vista topológico e da teoria da medida, seguindo os trabalhos de Michael Herman. A cada homeomorfismo do círculo podemos associar um número real positivo, denominado número de rotação. Mostraremos que existe um conjunto de números irracionais de medida de Lebesgue total na reta tal que, se f é um difeomorfismo do círculo de classe \'C POT. r \' que preserva a orientação, com r maior ou igual a 3 e com número de rotação nesse conjunto, então f é pelo menos \'C POT. r - 2\' -conjugada a uma translação irracional. Além disso, mostraremos que dado um caminho \'f IND. t\' de classe \'C POT. 1\' definido em um intervalo [a;b] no conjunto dos difeomorfismos do círculo de classe \'C POT. r\' que preservam a orientação, com r maior ou igual a 3, o conjunto dos parâmetros em que \'f IND. t\' é \'C POT. r - 2\' -conjugada a uma translação irracional tem medida de Lebesgue positiva, desde que os números de rotação em \'f IND. a\' e \'f IND. b\' sejam distintos / We will study the generic behavior of circle diffeomorphisms, in the combinatorial, topological and measure-theoretical sense, following the work of Michael Herman. To each order preserving homeomorphism of the circle we can associate a positive real number, called rotation number, which is invariant under conjugacy. We will show that there is a set of irrational numbers with full Lebesgue measure on R such that, if f is a circle diffeomorphism of class \'C POT. r\', with r greater or equal 3 and with rotation number in that set, then f is at least \'C POT. r - 2\' -conjugated to an irrational translation. Moreover, we will show that if ft is a \'C POT. 1\' -path defined on a interval [a;b] over the set of the circle diffeomorphisms orientation preserving, with r \'> or =\' 3, then the set of parameters where \'f IND. t\' is \'C POT. r - 2\' -conjugated to a irrational translation has positive Lebesgue measure, since the rotation numbers of \'f IND. a\' and \'f IND. b\' are distinct

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.