Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. 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.

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This thesis studies the large scale dimension theory of metric spaces. Background on dimension theory is provided, including topological and asymptotic dimension, and notions of nonpositive curvature in metric spaces are reviewed. The hyperbolic dimension of Buyalo and Schroeder is surveyed. Miscellaneous new results on hyperbolic dimension are proved, including a union theorem, an estimate for central group extensions, and the vanishing of hyperbolic dimension for countable abelian groups. A new quasi-isometry invariant called weak hyperbolic dimension (abbreviated $\wdim$) is introduced and developed. Weak hyperbolic dimension is computed for a variety of metric spaces, including the fundamental computation $\wdim \Hyp^n = n-1$. An estimate is proved for (not necessarily central) group extensions. Weak dimension is used to obtain the quasi-isometric nonembedding result $\Hyp^4 \not \rightarrow \Sol \times \Sol$ and possible directions for further nonembedding applications are explored. / Thesis / Doctor of Philosophy (PhD) / Shapes and spaces are studied from the "large scale" or "far away" point of view. Various notions of dimension for such spaces are studied.

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Gromov's theorem on finitely generated groups of polynomial growth is one of the cornerstones of geometric group theory. It has seen many applications and has immense importance within its realm. However, the early proofs of Gromov's theorem were based on deep results and were practically unsuitable for teaching. Recently, a new family of proofs have been published, potentially providing new ways to learn and teach Gromov's theorem. In this Master's thesis, we aim to provide an expository introduction to Gromov's theorem and its proofs.  We will first explore a brief theoretical overview of geometric group theory, which will include a proof of a theorem of Bass-Guivarc'h. We will then give a detailed account of an elementary proof of Gromov's theorem for groups of sub-polynomial growth, which was originally outlined by Tao in a blog post. The main contribution we make in this thesis is providing a significant amount of additional detail, bridging the gaps, and providing theoretical clarifications to all the steps in this proof. Finally, we will present some applications of Gromov's theorem. / Gromovs sats om ändligt genererade grupper med polynomiell tillväxt är en av grundpelarna inom geometrisk gruppteori. Satsen har sett många tillämpningar och har enorm betydelse inom sitt område. Dock baserades de tidiga bevisen för Gromovs sats på djupa resultat och var praktiskt taget olämpliga för undervisning. Nyligen har en ny familj av bevis publicerats, vilket potentiellt ger nya sätt att lära sig och undervisa Gromovs sats. I denna uppsats syftar vi till att ge en förklarande introduktion till Gromovs sats och dess bevis. Först ges en kort teoretisk översikt över geometrisk gruppteori, vilket kommer att inkludera ett bevis av en sats av Bass-Guivarc'h. Sedan ger vi en detaljerad redogörelse för ett elementärt bevis av Gromovs sats om grupper med sub-polynomiell tillväxt, vilket ursprungligen beskrevs av Tao i en bloggpost. Det huvudsakliga bidraget vi gör i denna uppsats är att tillhandahålla en betydande mängd ytterliga detaljer, och att ge teoretiska förtydliganden i alla steg i detta bevis. Slutligen presenterar vi några tillämpningar av Gromovs sats.

In 1911, mathematician Max Dehn posed three decision problems for finitely presented groups that have remained central to the study of combinatorial group theory. His work provided the foundation for geometric group theory, which aims to analyze groups using the topological and geometric properties of the spaces they act on. In this thesis, we study group actions on Cayley graphs and the Farey tree. We prove that a group has a solvable word problem if and only if its associated Cayley graph is constructible. Moreover, we prove that a group is finitely generated if and only if it acts geometrically on a proper path-connected metric space. As an example, we show that SL(2, Z) is finitely generated by proving that it acts geometrically on the Farey tree.

In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding sought-after group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups G(^m) and G(ⁿ) both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W(²) and W(³) are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. We construct a census of two-generator one-relator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of one-relator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the non-CAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. For every prime p we construct finite p-groups of arbitrary negative deficiency, and thereby complete Kotschick's proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients. We determine examples of sizeable graphs, as required to construct finitely presented non-hyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.

Cette thèse est consacrée à l'étude du groupe d'automorphismes de groupes agissant sur des arbres d'une part, et du rang des groupes de Coxeter d'autre part.<p><p>Via la théorie de Bass-Serre, un groupe agissant sur un arbre est doté d'une structure algébrique particulière, généralisant produits amalgamés et extensions HNN. Le groupe est en fait déterminé par certaines données combinatoires découlant de cette action, appelées graphes de groupes. <p><p>Un cas particulier de cette situation est celle d'un produit libre. Une présentation du groupe d'automorphisme d'un produit libre d'un nombre fini de groupes librement indécomposables en termes de présentation des facteurs et de leurs groupes d'automorphismes a été donnée par Fouxe-Rabinovich. Il découle de son travail que si les facteurs et leurs groupes d'automorphismes sont de présentation finie, alors le groupe d'automorphisme du produit libre est de présentation finie. Une première partie de cette thèse donne une nouvelle preuve de ce résultat, se basant sur le langage des actions de groupes sur les arbres.<p><p>Un groupe accessible est un groupe de type fini déterminé par un graphe de groupe fini dont les groupes d'arêtes sont finis et les groupes de sommets ont au plus un bout, c'est-à-dire qu'ils ne se décomposent pas en produit amalgamé ni en extension HNN sur un groupe fini. L'étude du groupe d'automorphisme d'un groupe accessible est ramenée à l'étude de groupes d'automorphismes de produits libres, de groupes de twists de Dehn et de groupes d'automorphismes relatifs des groupes de sommets. En particulier, on déduit un critère naturel pour que le groupe d'automorphismes d'un groupe accessible soit de présentation finie, et on donne une caractérisation des groupes accessibles dont le groupe d'automorphisme externe est fini. Appliqués aux groupes hyperboliques de Gromov, ces résultats permettent d'affirmer que le groupe d'automorphismes d'un groupe hyperbolique est de présentation finie, et donnent une caractérisation précise des groupes hyperboliques dont le groupe d'automorphisme externe est fini.<p><p>Enfin, on étudie le rang des groupes de Coxeter, c'est-à-dire le cardinal minimal d'un ensemble générateur pour un groupe de Coxeter donné. Plus précisément, on montre que si les composantes de la matrice de Coxeter déterminant un groupe de Coxeter sont suffisamment grandes, alors l'ensemble générateur standard est de cardinal minimal parmi tous les ensembles générateurs. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished