Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [51]-52).

Thesis (M.A.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2009. / Includes bibliographical references.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 94-98).

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2006. / This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.

Made available in DSpace on 2014-06-11T19:27:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-10-08Bitstream added on 2014-06-13T20:27:37Z : No. of bitstreams: 1 santos_wr_me_rcla.pdf: 678629 bytes, checksum: 2360a8945944381828cffbf939ac6872 (MD5) / O objetivo deste trabalho é apresentar a teoria elementar de grupos, segundo uma abordagem geométrica. Apresentamos uma introdução aos grupos de simetrias de sólidos regulares e como aplicação apresentamos os sete grupos de frisos / In this work we present a geometric approach to the study of elementary group theory. We give an introduction to symmetry groups of regular solids and as an application we present the seven Frieze groups

The Thompson groups $F, T$ and $V$ are important groups in geometric group theory: $T$ and $V$ being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, for $n$ and $r$ natural numbers and $1 < r < n$, the groups $F_{n}$, $T_{n,r}$ and $G_{n,r}$ ($T ≅ T_{2,1}$ and $V ≅ G_{2,1}$). Automorphisms of $F$ and $T$ were characterised in the seminal paper of Brin ([16]) and, later on, Brin and Guzman ([17]) investigate automorphisms of $T_{n, n-1}$ and $F_{n}$ for $n > 2$. However, their techniques give no information about automorphisms of $G_{n,r}$. The second chapter of this thesis is dedicated to characterising the automorphisms of $G_{n,r}$. Presenting results of the author's article [10], we show that automorphisms of $G_{n,r}$ are homeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition. In the rest of Chapter 2 and early sections of Chapter 3 we investigate the group $\out{G_{n,r}}$ of outer automorphisms of $G_{n,r}$. Presenting results of the forthcoming article [6] of the author's, we show that there is a subgroup $\hn{n}$ of $\out{G_{n,r}}$, independent of $r$, which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem in $\hn{n}$ and is based on [44]. We give necessary and sufficient conditions for an element of $\hn{n}$ to have finite order, although these do not yield a decision procedure. Given an automorphism $\phi$ of a group $G$, two elements $f, g ∈ G$ are said to be $\phi$-twisted conjugate to one another if for some $h ∈ G$, $g = h−1 f (h)\phi$. This defines an equivalence relation on $G$ and $G$ is said to have the $\rfty$ property if it has infinitely many $\phi$-twisted conjugacy classes for all automorphisms $\phi ∈ \aut{G}$. In the final chapter we show, using the description of $\aut{G_{n,r}}$, that for certain automorphisms, $G_{n,r}$ has infinitely many twisted conjugacy classes. We also show that for certain $\phi ∈ \aut{G_{2,1}}$ the problem of deciding when two elements of $G_{2,1}$ are $\phi$-twisted conjugate to one another is soluble.

Orientador: Elíris Cristina Rizziolli / Banca: Daiane Alice Henrique Ament / Banca: Thiago de Melo / Resumo: Neste trabalho tratamos da classificação dos grupos de friso. Para realizar este objetivo abordamos elementos básicos da estrutura algébrica de grupo bem como apresentamos transformações geométricas, entre estas destacamos: translações, reflexões, rotações e reflexão com deslizamento. Além disso, localizamos este assunto como tópico da estrutura curricular do Ensino Fundamental e executamos uma atividade em sala de aula em que os alunos criaram frisos ornamentais / Abstract: In this work we deal with the classification of frieze groups. In order to accomplish this objective we approach basic elements of the algebraic group structure as well as present geometric transformations, among which we highlight: translations, reflections, rotations and glide reflection. In addition, we locate this subject as a topic of the curricular structure of Elementary School and perform a classroom activity in which the students created ornamental friezes / Mestre

Made available in DSpace on 2014-10-09T12:29:05Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:03:10Z (GMT). No. of bitstreams: 1 01003.pdf: 2868809 bytes, checksum: b311b542d0e54ce8f0ce8ed1e274b64c (MD5) / Dissertacao (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP