Möbius-like groups of homeomorphisms of the circle

A group G → Homeo_+(S^1) is a Möbius-like group if every element of G is conjugate in Homeo(S^1) to a Mobius transformation. Our main result is: given a Mobus like like group G which has at least one global fixed point, G is conjugate in Homeo(S^1) to a Möbius group if and only if the limit set of G is all of S^1 . Moreover, we prove that if the limit set of G is not SI, then after identifying some closed subintervals of S^1 to points, the induced action of G is conjugate to an action of a Möbius group.

We also show that the above result does not hold in the case when G has no global fixed points. Namely, we construct examples of Möbius-like groups with limit set equal to S^1, but these groups cannot be conjugated to Möbius groups.

Descriptive set theory and the ergodic theory of countable groups

The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Abért and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C^*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.

Class two p groups as fixed point free automorphism groups

Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If p^c ≠ r^d + 1 for any c = 1, 2 and any prime r where r^2d+1 divides |G| and if C_G(A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|.

The theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A₁, a subgroup of A, where A₁ centralizes D(R), then all irreducible characters of A₁R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed.

Graphs and Groups

This project is a combination of graphs and group theory in which the aim is to describe the automorphism group of some specific families of graphs. Finally, an example of the application of automorphism groups in reaction graphs is shown. The project is written in english.

Normal structures and automorphism groups of t-designs

Combinatorial configurations known as t-designs are studied. These are pairs ˂B, ∏˃, where each element of B is a k-subset of ∏, and each t-design occurs in exactly λ elements of B, for some fixed integers k and λ. A theory of internal structure of t-designs is developed, and it is shown that any t-design can be decomposed in a natural fashion into a sequence of “simple” subdesigns. The theory is quite similar to the analysis of a group with respect to its normal subgroups, quotient groups, and homomorphisms. The analogous concepts of normal subdesigns, quotient designs, and design homomorphisms are all defined and used.

This structure theory is then applied to the class of t-designs whose automorphism groups are transitive on sets of t points. It is shown that if G is a permutation group transitive on sets of t letters and ф is any set of letters, then images of ф under G form a t-design whose parameters may be calculated from the group G. Such groups are discussed, especially for the case t = 2, and the normal structure of such designs is considered. Theorem 2.2.12 gives necessary and sufficient conditions for a t-design to be simple, purely in terms of the automorphism group of the design. Some constructions are given.

Finally, 2-designs with k = 3 and λ = 2 are considered in detail. These designs are first considered in general, with examples illustrating some of the configurations which can arise. Then an attempt is made to classify all such designs with an automorphism group transitive on pairs of points. Many cases are eliminated of reduced to combinations of Steiner triple systems. In the remaining cases, the simple designs are determined to consist of one infinite class and one exceptional case.

Effect of hydroxyl groups on nonradiative decay of Er3+: I-4(13/2)-> I-4(15/2) transition in zinc tellurite glasses

A series of zinc tellurite glasses of 75TeO(2)-20ZnO-(5-x)La2O3-xEr(2)O(3) (x=0.02, 0.05, and 0.1 mol%) with the different hydroxl groups were prepared by the conventional melt-quenching method. Infrared spectra were measured in order to estimate the exact content of OH- groups in samples. The observed increase of the fluorescence lifetime with the oxygen bubbling time has been related to the reduction in the OH- content concentration as evidenced by IR transmission spectra. Various nonradiative decay rates from I-4(13/2) of Er3+ with. the change of OH content were determined from the fluorescence lifetime and radiative decay rates were calculated on the basis of Judd-Ofelt theory. (c) 2005 Elsevier B.V. All rights reserved.

The effect of OH- groups on the spectroscopic properties of erbium-doped tellurite glasses

A series of five different concentration erbium-doped tellurite glasses with various hydroxl groups were prepared. Infrared spectra of glasses were measured. In order to estimate the exact content of OH- groups in samples, various absorption coefficients of the OH- vibration band were analyzed under the different oxygen bubbling times. The absorption spectra of the glasses were measured, and the Judd-Ofelt intensity parameters Omega(i) of samples with the different erbium ions concentration and OH- contents were calculated on the basis of the Judd-Ofelt theory. The peak stimulated emission cross-section of (I13/2 ->I15/2)-I-4-I-4 transition of the samples was finally calculated by using the McCumber theory. The fluorescence spectra of Er3+:I-4(13/2)->I-4(15/2) transition and the lifetime of Er3+:I-4(13/2) level of the samples were measured. The effects of OH- groups on the spectroscopic properties of Er3+ doped samples with the different concentrations were discussed. The results showed that the OH- groups had great influences on the Er3+ lifetime and the fluorescence peak intensity. The OH- group is a main influence factor of fluorescence quenching when the doping concentration of Er2O3 is smaller than 1.0 mol%, but higher after this concentration, the energy transfer of Er3+ ions turns into the main function of the fluorescence quenching. And basically, there is no influence on the other spectroscopic properties (FWHM, absorption spectra, peak stimulated emission cross section, etc.).

The influence of OH groups on laser performance in phosphate glasses

Because of the influence of OH groups in phosphate glasses on the radiation of rare-earth ions, the laser performance is degraded. The laser efficiency and the small signal gain experiment of several phosphate glass samples have been done, the concentration of OH groups in glasses was calculated from the measured absorption coefficient at 3.47 μm. It is shown that the concentration of OH groups in phosphate glasses can seriously influence the laser output characteristics, and the OH groups have worse influence on the laser amplifier than laser oscillator.