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Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-22T13:40:01Z No. of bitstreams: 2 Dissertação - Tulio Marcio Gentil dos Santos - 2017.pdf: 1903129 bytes, checksum: 68678e5a2933f0e40216c1e1181aa7bc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-22T13:40:22Z (GMT) No. of bitstreams: 2 Dissertação - Tulio Marcio Gentil dos Santos - 2017.pdf: 1903129 bytes, checksum: 68678e5a2933f0e40216c1e1181aa7bc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-22T13:40:22Z (GMT). No. of bitstreams: 2 Dissertação - Tulio Marcio Gentil dos Santos - 2017.pdf: 1903129 bytes, checksum: 68678e5a2933f0e40216c1e1181aa7bc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Let $F_d$ be the free group of rank $d$, freely generated by $\{y_1,...,y_d\}$, $\mathbb{D}F_d$ the group ring over an integral domain $\mathbb{D}$, $E_d$ subset of $F_d$ containing $\{y_1,...,y_d\}$, $p_s(x)=x^n+c_{s,n-1}x^{n-1}+...+c_{s,1}x+c_{s,0} \in \mathbb{D}[x]$ a monic polynomial and the quotient ring $$A(d,n,E_d)=\frac{\mathbb{D}F_d}{\langle p_s(s):s\in E_d \rangle_{ideal}}.$$ When $p_s(s)$ is cubic for all $s$, we construct a finite set $E_d$ such that $A(d,n,E_d)$ has finite rank over an extension of $\mathbb{D}$. In the case where all polynomials are equal to $(x-1)^3$ and $\mathbb{D}=\mathbb{Z}[\frac{1}{6}]$ we construct a finite subset $P_d$ of $F_d$ such that $A(d,3,P_d)$ has finite $\mathbb{D}$-rank and its augmentation ideal is nilponte. Furthermore $(x-1)^3$is satisfied by all elements in the image of $F_2$ in $A(2,3,P_2)$. / Sejam $F_d$ um grupo livre de posto $d$, livremente gerado por $\{y_1,...,y_d\}$, $\mathbb{D}F_d$ o anel de grupo sobre o domínio de integridade $\mathbb{D}$, $E_d$ subconjunto de $F_d$ contendo $\{y_1,...,y_d\}$, $p_s(x)=x^n+c_{s,n-1}x^{n-1}+...+c_{s,1}x+c_{s,0} \in \mathbb{D}[x]$ e o anel quociente $$A(d,n,E_d)=\frac{\mathbb{D}F_d}{\langle p_s(s):s\in E_d \rangle_{ideal}}.$$ Quando $p_s(s)$ é cúbico para todo $s$, construímos um conjunto finito $E_d$ tal que $A(d,n,E_d)$ tem posto finito sobre uma extensão de $\mathbb{D}$. No caso em que todos os polinômios são iguais a $(x-1)^3$ e $\mathbb{D}=\mathbb{Z}[\frac{1}{6}]$, construímos um subconjunto finito $P_d$ de $F_d$ tal que $A(d,3,P_d)$ tem $\mathbb{D}$-posto finito e seu ideal de aumento é nilpotente. Além disso $(x-1)^3$ é satisfeita por todos elementos na imagem de $F_2$ em $A(2,3,P_2)$.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24383号 / 理博第4882号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 葉廣 和夫, 教授 加藤 毅, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This thesis discusses the Kaplansky zero divisor conjecture. The conjecture states that a group ring of a torsion-free group over a field has no nonzero zero divisors. There are situations for which this conjecture is known to hold, such as linearly orderable groups, unique product groups, solvable groups, and elementary amenable groups. This paper considers the possibility that the conjecture is false and there is some counterexample in existence. The approach to searching for such a counterexample discussed here is to first find a torsion-free group that has subsets A and B such that AB has no unique product. We do this by exhaustively searching for the subsets A and B with fixed small sizes. When |A| = 1 or 2 and |B| is arbitrary we know that AB contains a unique product, but when |A| is larger, not much was previously known. After an example is found we then verify that the sets are contained in a torsion-free group and further investigate whether the group ring yields a nonzero zero divisor. Together with Dr. Pace P. Nielsen, assistant math professor of Brigham Young University, we created code that was implemented in Magma, a computational algebra system, for the purpose of considering each size of A and B and running through each case. Along the way we check for the possibility of torsion elements and for other conditions that lead to contradictions, such as a decrease in the size of A or B. Our results are the following: If A and B are sets of the sizes below contained in a torsion-free group, then they must contain a unique product. |A| = 3 and |B| ≤ 16; |A| = 4 and |B| ≤ 12; |A| = 5 and |B| ≤ 9; |A| = 6 and |B| ≤ 7. We have continued to run cases of larger size and hope to increase the size of B for each size of A. Additionally, we found a torsion-free group containing sets A and B, both of size 8, where AB has no unique product. Though this group does not yield a counterexample for the Kaplansky zero divisor conjecture, it is the smallest explicit example of a non-uniqueproduct group in terms of the size of A and B.

For a oriented genus g surface with one boundary component, S_g, the Torelli group is the group of orientation preserving homeomorphisms of S_g that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F'' where F=π_1(S_g) and F'' is the second term of the derived series. I show that the kernel of the Magnus representation, Mag(S_g), is highly non-trivial and has a rich structure as a group. Specifically, I define an infinite filtration of Mag(S_g) by subgroups, called the higher order Magnus subgroups, M_k(S_g). I develop methods for generating nontrivial mapping classes in M_k(S_g) for all k and g≥2. I show that for each k the quotient M_k(S_g)/M_{k+1}(S_g) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally I show that for g≥3 the quotient M_k(S_g)/M_{k+1}(S_g) surjects onto an infinite rank torsion free abelian group. To do this, I define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.

Orientador: Dessislava H. Kochloukova / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T20:51:24Z (GMT). No. of bitstreams: 1 Norte_ElianaVieira_M.pdf: 1155254 bytes, checksum: cc4bea0f77a79d67e969652289584764 (MD5) Previous issue date: 2006 / Resumo: Nessa dissertação estudamos a teoria de Bass-Serre que liga grupos que agem sobre árvores em grupos fundamentais de grafos de grupos. Para desenvolver essa teoria primeiramente estudamos conceitos básicos como: grupos livres, produto livre amalgamado, extensão HNN. Na parte final a teria de Bass-Serre é aplicada para endomorfismos injetivos de grupos de Baumslag-Solitar / Abstract: ln this master thesis we study Bass-Serre theory that links groups acting on trees and fundamental groups of graph of groups. To develop the theory we study first basic concepts as: free groups, free product with amalgamation, HNN extension. At the end Bass-Serre theory is used to study the injective endomorphisms of Baumslag-Solitar groups / Mestrado / Algebra / Mestre em Matemática

Orientador: Dessislava Hristova Kochloukova / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:11:38Z (GMT). No. of bitstreams: 1 Araujo_PaulaMacedoLinsde_M.pdf: 989128 bytes, checksum: 16ef5d7a0a914714bc4163e7549b1c3a (MD5) Previous issue date: 2014 / Resumo: Estudamos um resultado que se encontra no artigo "Finitely Presented Wreath Products And Double Coset Decompositions" de Y. de Cornulier que afirma que o produto entrelaçado entre os grupos W e G, com respeito a um G-conjunto X, é finitamente apresentável se, e somente se, as seguintes condições são satisfeitas: i. W e G são finitamente apresentáveis; ii. G age sobre X com estabilizadores finitamente gerados; iii. G age diagonalmente sobre X x X com finitas órbitas / Abstract: We study a result in the paper "Finitely Presented Wreath Products And Double Coset Decompositions" by Y. de Cornulier, which asserts that the wreath between the groups W and G with respect to a G-set X is finitely presented if and only if the following conditions hold: i. W and G are finitely presented; ii. G acts on X with finitely generated stabilizers; iii. G acts diagonally on X x X with finitely many orbits / Mestrado / Matematica / Mestra em Matemática

Let F be a free group of finite rank. Given words u, v ∈ F, J.H.C. Whitehead solved the decision problem of finding an automorphism φ ∈ Aut(F), carrying u to v. He used topological methods to produce an algorithm. Higgins and Lyndon gave a very concise proof of the problem based on the works of Rapaport. We provide a detailed account of Higgins and Lyndon’s proof of the peak reduction lemma and the restricted version of Whitehead’s theorem, for cyclic words as well as for sets of cyclic words, with a full explanation of each step. Then, we give an inductive proof of Whitehead’s minimization theorem and describe Whitehead’s decision algorithm. Noticing that Higgins and Lyndon’s work is limited to the cyclic words, we extend their proofs to ordinary words and sets of ordinary words. In the last chapter, we mention an example given by Whitehead to show that the decision problem for finitely generated subgroups is more difficult and outline an approach due to Gersten to overcome this difficulty. We also give an extensive literature survey of Whitehead’s algorithm / Thesis / Master of Science (MSc)

The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free group and if we take all possible relations we have a free abelian group. We show that if no finite index subgroup of a group $G$ contains a normal subgroup that maps onto $mathbb{Z}$, then every homomorphism from $G$ to the outer automorphism group of a free group has finite image. The above criterion is satisfied by SL$_m(mathbb{Z})$ for $m geq 3$ and, more generally, all irreducible lattices in higher-rank, semisimple Lie groups with finite centre. Given a right-angled Artin group $A_Gamma$ we find an integer $n$, which may be easily read off from the presentation of $A_G$, such that if $m geq 3$ then SL$_m(mathbb{Z})$ is a subgroup of the outer automorphism group of $A_Gamma$ if and only if $m leq n$. More generally, we find criteria to prevent a group from having a homomorphism to the outer automorphism group of $A_Gamma$ with infinite image, and apply this to a large number of irreducible lattices as above. We study the subgroup $IA(A_Gamma)$ of $Aut(A_Gamma)$ that acts trivially on the abelianisation of $A_Gamma$. We show that $IA(A_Gamma)$ is residually torsion-free nilpotent and describe its abelianisation. This is complemented by a survey of previous results concerning the lower central series of $A_Gamma$. One of the commonly used generating sets of $Aut(F_n)$ is the set of Whitehead automorphisms. We describe a geometric method for decomposing an element of $Aut(F_n)$ as a product of Whitehead automorphisms via Stallings' folds. We finish with a brief discussion of the action of $Out(F_n)$ on Culler and Vogtmann's Outer Space. In particular we describe translation lengths of elements with regards to the `non-symmetric Lipschitz metric' on Outer Space.

Neste trabalho apresentamos uma descrição geométrica do grupo de tranças no disco Bpnq e sua apresentação em termos de geradores e relatores no famoso teorema da apresentação de Artin. Mostraremos também que o grupo de tranças puras PBpnq, grupo que possui a permutação trivial das cordas, é bi-ordenável, ou seja, exibiremos uma ordenação para PBpnq que será invariante pela multiplicação em ambos os lados. Esse processo é dado a partir da combinação da técnica de pentear Artin e a expansão Magnus para grupos livres. / In this work, we present a geometric description of the braids groups of the disk Bpnq and its presentation in terms of generators and relations in the famous theorem of Artin\'s presentation. We also show that groups of pure braids, denoted by PBpnq, groups that have the trivial permutation of the strings, are bi-orderable, that is, we will present the explicit construction of a strict total ordering of pure braids PBpnq which is invariant under multiplying on both sides. This process is given from the combination of the techniques of combing Artin and Magnus expansion to free groups.