Wo, Weifeng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 99-102). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

by Wong Chun-fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 81-82). / Chapter CHAPTER0 --- Introduction --- p.1 / Chapter CHAPTER1 --- Teichmuller Space of genus g --- p.5 / Chapter 1.1. --- Teichmiiller Space of genus g / Chapter 1.2. --- Fuchsian Model and Discrete subgroup of Aut(H) / Chapter 1.3. --- Fricke Space / Chapter CHAPTER2 --- Hyperbolic Geometry and Fenchel-Nielsen Coordinates --- p.14 / Chapter 2.1. --- Poincare Metric and Hyperbolic Geometry / Chapter 2.2. --- Fenchel-Nielsen Coordinates / Chapter 2.3. --- Fricke-Klein Embedding / Chapter CHAPTER3 --- Quasiconformal Mappings --- p.23 / Chapter 3.1. --- Definitions / Chapter 3.2. --- Existence Theorems on Quasiconformal Mappings / Chapter 3.3. --- Dependence on Beltrami Coefficients / Chapter CHAPTER4 --- Teichmuller Spaces --- p.37 / Chapter 4.1. --- Analytic Construction of Teichmiiller Spaces / Chapter 4.2. --- Teichmiiller mapping and Teichmiiller Theorem / Chapter 4.3. --- Teichmiiller Uniqueness Theorem / Chapter CHAPTER5 --- Complex Analytic Theory of Teichmiiller Spaces --- p.50 / Chapter 5.1. --- Bers' Embedding and the complex structure of Teichmiiller Space / Chapter 5.2. --- Invariance of Complex Structure of Teichmiiller Space / Chapter 5.3. --- Teichmiiller Modular Groups / Chapter 5.4. --- Classification of Teichmiiller Modular Transformations / Chapter CHAPTER6 --- Weil-Petersson Metric --- p.68 / Chapter 6.1. --- Petersson Scalar Product and Reproducing formula / Chapter 6.2. --- Infinitesimal Theory of Teichmuller Spaces / Chapter 6.3. --- Weil-Petersson Metric / BIBLIOGRAPHY --- p.81

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 119-121. / Ph.D.

Ke, Ting. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 30-32). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction and Main results --- p.1 / Chapter 2 --- Preliminaries --- p.7 / Chapter 3 --- Finite Speed of Propagation Property --- p.11 / Chapter 4 --- Proof of the Main Results --- p.19 / Chapter 4.1 --- Proof of Theorem 1.0.1 --- p.19 / Chapter 4.2 --- Proof of Theorem 1.0.2 --- p.24 / Chapter 5 --- Discussions --- p.26 / Bibliography --- p.30

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.

abstract: In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1). / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

This work is a compilation of structural results for the von Neumann algebras of poly-hyperbolic groups established in a series of works done jointly with I. Chifan and T. Sinclair; and S. Pant. These works provide a wide range of circumstances where the product structure, a discrete structural property, can be recovered from the von Neumann algebra (a continuous object). The primary result of Chifan, Sinclair and myself is as follows: if Γ = Γ1 × · · · × Γn is a product of non-elementary hyperbolic icc groups and Λ is a group such that L(Γ)=L(Λ), then Λ decomposes as an n-fold product of infinite groups. This provides a group-level strengthening of the unique prime decomposition of Ozawa and Popa by eliminating any assumption on the target group Λ. The methods necessary to establish this result provide a malleable procedure which allows one to rebuild the product of a group from the algebra itself. Modifying the techniques found in the previous work, Pant and I are able to demonstrate that the class of poly-groups exhibit a similar phenomenon. Specifically, if Γ is a poly-hyperbolic group whose corresponding algebra is non-prime, then the group must necessarily decompose as a product of infinite groups.

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be linear fractional transformations when in $U^2$ and $B^2$. We will then study a bit of topology, specifically the topological groups relevant to the group of isometries of hyperbolic $n$-space, $I(H^n)$. Finally we will combine what we know about hyperbolic $n$-space and topological groups in order to study fundamental regions, fundamental domains, Dirichlet domains, and quotient spaces. Using examples in $U^2$, we will then illustrate how useful fundamental domains are when it comes to visualizing the geometry of quotient spaces.

Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [10]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [59]. We also obtain a product version of this result: any maximal abelian ∗-subalgebra of any II 1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana's cocycle superrigidity theorem [36], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article [11] which has already been submitted for publication.