We use profinite Bass-Serre theory (the theory of profinite group actions on profinite trees) to prove that the fundamental groups of finite graphs of free groups which are l-acylindrical and have finitely generated edge groups are conjugacy separable. We apply this theorem to: demonstrate that a generic positive one-relator group is conjugacy separable; produce a variant of the Rips con- struction in which the output group is conjugacy separable; apply this last to exhibit an example of a strong profinite equivalence between two finitely presented groups, one of which is conjugacy separable and the other having unsolvable conjugacy problem. We further use profinite Bass-Serre theory to demonstrate that having one end is an up-weak pro-C property for any extension- closed class C of finite groups. We show by example that it is not a down-weak pro-p property for any prime p. We consider Korenev's definition of pro-p ends for a pro-p group, and show that the number of ends of a finitely generated residually p group cannot be less than the number of pro-p ends of its pro-p completion. We explore possibilities for, but are ultimately unsuc- cessful in giving, a proper analogue of Stallings' theorem for pro-p groups. We ask which other properties might be profinite, and use another variant of the Rips construction to produce examples of patholog- ical groups such that either they are hyperbolic groups which are not residually finite, or neither property (FA) nor property (T) is an up-weak profinite property.

O entendimento de quebra espontânea de simetria é um problema importante para o estudo de fenômenos na evolução de sistemas abertos, tanto em física quanto em química, como também na biologia. Aqui estudamos um método a mais para este tipo de análise, usando conceitos de sistemas dinâmicos com simetria. O sistema dinâmico escolhido é discreto, isto é, realizado por iteração de um difeomorfismo equivariante sob a ação de um grupo compacto, neste caso um grupo finito do tipo Dm. Especificamente, investigamos o comportamento de atratores caóticos sob a variação dos parâmetros. / The understanding of spontaneous symmetry breaking is an important problem in the study of phenomena in the evolution of open systems, in physics and chemistry as well as in biology. Here we study another method for this kind of analysis, using concepts from dynamical systems with symmetry. The chosen dynamical system is discrete, that is, realized by iteration of an equivariant diffeomorphism under the action of a compact group, in this case one of the finite groups of type Dm. Specifically, we investigate the behavior of chaotic attractors under variation of the parameters.

Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.

by Guan-xin Li. / "January 1999." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 223-225). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Ai Jun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 100-103). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

In this thesis, we prove that local genericity implies globally genericity for the quasi-split unitary group U3 for a quadratic extension of number fields E/F. We follow [Fli1992] and [GJR2001] closely, using the relative trace formula approach. Our main result is the existence of smooth transfer for the relative trace formulae in [GJR2001], which is circumvented there. The basic idea is to compute the Mellin transform of Shalika germ functions and show that they are equal in the unitary case and the general linear case.

分形分析的主題是研究分形上的Dirichlet形式和Laplacian. 壓縮的自相似群有一個與之關聯的極限空間,此空間通常具備分形結構，因而引發了分形分析和自相似群兩個分支的結合. / 我們回顧了自相似群和它們的極限空間極限空間可以用Schreier 圖來逼近,事實上其可以看成由Schreier圖構造出來的雙曲圖的雙曲邊界.我們探究了迭代單值群. 通過增加專門的條件我們可以得到迭代單值群的極限空間同胚於某個Julia集. / 通過運用[31] 中的想法和[47] 中自相似隨機游動的方法，我們闡明了極限空間上Laplacian和Dirichlet形式的構造步驟我們介紹了加法器， Basilica群以及Hanoi塔群的極限空間(在第三種情況下是Sierpiríski墊片)上的Laplacian 這裡得到的Dirichlet形式是局部且正則的. / 通過採用[53] 的設置, 我們描述了加法器的極限空間上的誘發型Dirichlet形式在構造了加法器的自相似圖上的嚴格可逆隨機游動後，我們可以得到一個非局部的Dirichlet形式. / The major theme of fractal analysis is studying Dirichlet forms and Laplacians on fractals. For a contracting self-similar group there is an associated limit space, which usually exhibits a fractal structure, thereby triggering the combination of fractal analysis and self-similar groups. / We give reviews of self-similar groups and their limit spaces. Limit space can be approximated by Schreier graphs, and it is in fact identied as a hyperbolic boundary of a hyperbolic graph constructed from Schreier graphs. We explore the iterated monodromy groups. By adding technical conditions, we have that the limit space of an iterated monodromy group is homeomorphic to a Julia set. / We show the construction process of Laplacians and Dirichlet forms on limit spaces using the idea of [31] and the method of self-similar random walks from [47]. We present examples of Laplacians of the limit spaces of adding machine, the Basilica group and the Hanoi Tower group (it is Sierpi´nski gasket in this case). In this context these forms are local and regular. / We describe the induced Dirichlet forms on limit space of the adding machine by adopting the settings of [53] . By constructing strictly reversible random walks on self-similarity graph of the adding machine, we can obtain a non-local Dirichlet form. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lin, Dateng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 71-76). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Review of fractal analysis --- p.6 / Chapter 1.2 --- Applications to self-similar groups --- p.7 / Chapter 1.3 --- Boundary theory method --- p.8 / Chapter 1.4 --- Summary of the thesis --- p.9 / Chapter 2 --- Self-similar groups --- p.11 / Chapter 2.1 --- Basic definitions --- p.11 / Chapter 2.2 --- Limit spaces of self-similar groups --- p.18 / Chapter 2.3 --- Schreier graphs approximations --- p.24 / Chapter 2.4 --- Iterated monodromy groups --- p.28 / Chapter 3 --- Construction of Laplacians on limit spaces --- p.35 / Chapter 3.1 --- Dirichlet forms, Laplacians and resistance forms --- p.35 / Chapter 3.2 --- Representations of groups and functions --- p.42 / Chapter 3.3 --- Laplacians on limit spaces --- p.45 / Chapter 4 --- Induced Dirichlet form on limit space of the adding machine --- p.53 / Chapter 4.1 --- Martin boundary and hyperbolic boundary --- p.53 / Chapter 4.2 --- Graph energy and the induced form --- p.62 / Chapter 4.3 --- Induced Dirichlet form of the adding machine --- p.65 / Bibliography --- p.71

In general, a finitely presented group can have very nasty properties, but many of these properties are avoided if the group is assumed to admit a nice action by isometries on a space with a negative curvature property, such as Gromov hyperbolicity. Such groups are surprisingly common: there is a sense in which a random group admits such an action, as do some groups of classical interest, such as fundamental groups of closed Riemannian manifolds with negative sectional curvature. If a group admits an action on a Gromov hyperbolic space then large scale properties of the space give useful invariants of the group. One particularly natural large scale property used in this way is the Gromov boundary. The Gromov boundary of a hyperbolic group is a compact metric space that is, in a sense, approximated by spheres of large radius in the Cayley graph of the group. The technical results contained in this thesis are effective versions of this statement: we see that the presence of a particular topological feature in the boundary of a hyperbolic group is determined by the geometry of balls in the Cayley graph of radius bounded above by some known upper bound, and is therefore algorithmically detectable. Using these technical results one can prove that certain properties of a group can be computed from its presentation. In particular, we show that there are algorithms that, when given a presentation for a one-ended hyperbolic group, compute Bowditch's canonical decomposition of that group and determine whether or not that group is virtually Fuchsian. The final chapter of this thesis studies the problem of detecting Cech cohomological features in boundaries of hyperbolic groups. Epstein asked whether there is an algorithm that computes the Cech cohomology of the boundary of a given hyperbolic group. We answer Epstein's question in the affirmative for a restricted class of hyperbolic groups: those that are fundamental groups of graphs of free groups with cyclic edge groups. We also prove the computability of the Cech cohomology of a space with some similar properties to the boundary of a hyperbolic group: Otal's decomposition space associated to a line pattern in a free group.

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or a 5 x 4 puzzle, respectively. We consider the possibilities when we delete a color to make the game a 3 × 3 puzzle and when we add a color, making the game a 5 × 5 puzzle. Finally, we determine if solution two is a permutation of solution one.

In this thesis, we have presented our discovery of true finite homomorphic images of various permutation and monomial progenitors, such as 2*7: D14, 2*7 : (7 : 2), 2*6 : S3 x 2, 2*8: S4, 2*72: (32:(2S4)), and 11*2 :m D10. We have given delightful symmetric presentations and very nice permutation representations of these images which include, the Mathieu groups M11, M12, the 4-fold cover of the Mathieu group M22, 2 x L2(8), and L2(13). Moreover, we have given constructions, by using the technique of double coset enumeration, for some of the images, including M11 and M12. We have given proofs, either by hand or computer-based, of the isomorphism type of each image. In addition, we use Iwasawa's Lemma to prove that L2(13) over A5, L2(8) over D14, L2(13) over D14, L2(27) over 2D14, and M11 over 2S4 are simple groups. All of the work presented in this thesis is original to the best of our knowledge.