The Mobius function of the small Ree groups

The M\"obius function for a group, $G$, was introduced in 1936 by Hall in order to count ordered generating sets of $G$. In this paper we determine the M\"obius function of the simple small Ree groups, $R(q)={}^2G_2(q)$ where $q=3^{2m+1}$ for $m>0$, using their 2-transitive permutation representation of degree $q^3+1$ and describe their maximal subgroups in terms of this representation. We then use this to determine $\vert$Epi$(\Gamma,G)\vert$ for various $\Gamma$, such as $F_2$ or the modular group $PSL_2(\mathbb{Z})$, with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.Comment: Includes the determination of the M\"obius function for various finitely presented groups, such as $F_2$ and $PSL_2(\mathbb{Z})$ with applications to probabilistic generatio

Some calculations on the action of groups on surfaces

In this thesis we treat a number of topics related to generation of finite groups with motivation from their action on surfaces. The majority of our findings are presented in two chapters which can be read independently. The first deals with Beauville groups which are automorphism groups of the product of two Riemann surfaces with genus g > 1, subject to some further conditions. When these two surfaces are isomorphic and transposed by elements of G we say these groups are mixed, otherwise they are unmixed. We first examine the relationship between when an almost simple group and its socle are unmixed Beauville groups and then go on to determine explicit examples of several infinite families of mixed Beauville groups. In the second we determine the Mobius function of the small Ree groups 2G2(32m+1) = R(32m+1), where m >0, and use this to enumerate various ordered generating n-tuples of these groups. We then apply this to questions of the generation and asymptotic generation of the small Ree groups as well as interpretations in other categories, such as the number of regular coverings of a surface with a given fundamental group and whose covering group is isomorphic to R(32m+1)

Some calculations on the action of groups on surfaces

In this thesis we treat a number of topics related to generation of finite groups with motivation from their action on surfaces. The majority of our findings are presented in two chapters which can be read independently. The first deals with Beauville groups which are automorphism groups of the product of two Riemann surfaces with genus g > 1, subject to some further conditions. When these two surfaces are isomorphic and transposed by elements of G we say these groups are mixed, otherwise they are unmixed. We first examine the relationship between when an almost simple group and its socle are unmixed Beauville groups and then go on to determine explicit examples of several infinite families of mixed Beauville groups. In the second we determine the Mobius function of the small Ree groups 2G2(32m+1) = R(32m+1), where m >0, and use this to enumerate various ordered generating n-tuples of these groups. We then apply this to questions of the generation and asymptotic generation of the small Ree groups as well as interpretations in other categories, such as the number of regular coverings of a surface with a given fundamental group and whose covering group is isomorphic to R(32m+1)

The Free Statesman [March 9, 1967]

The Free Statesman, Volume 1, number 3, March 9, 196

Characterization of Cosmic Microwave Background Temperature Fluctuations for Cosmological Topologies Specified by Topological Betti Numbers

The cosmic microwave background explorer, COBE, and the Balloon Observations of Millimetric Extragalactic Radiation and Geophysics, BOOMERanG, have collected data from the universe and detected relic anisotropies indicative of one of the earliest events of the universe, decoupling. Hidden in the correlation between these temperature fluctuations is the signature of the global shape, or topology, of the early universe. It is possible to calculate the temperature fluctuations as due to primeval adiabatic density temperature fluctuations from the Sachs-Wolfe effect, which contains a topological term. Here we investigate some of the spaces and how they affect the microwave background. A large class of spaces can be understood with some tools of topology associated with the way curves and volumes divide a space. The three tools of interest are the Euler Characteristic, the Betti numbers and fundamental domains. We intend to demonstrate the relationship between topology and anisotropy by creating a general equation that contains a topological term based on a topological invariant, the Betti Numbers of a manifold, and a term based on the temperature fluctuations. We will also extend the work of Silk on classifying the finite flat spaces through the use of a new three-dimensional plotting technique. These graphs demonstrate the inadequacy of the current CMB data\u27s topological predictive properties. As many shapes can produce the observed peaks. We will also extend the work of Inoue on classifying compact hyperbolic manifolds in two ways. First we demonstrate that the compact hyperbolic eigenmodes can be represented by a pFq function, and do not need to be calculated individually. Secondly, we extend the power spectrum plot the high angular resolution, large l, and compare compact hyperbolic manifolds to the observed peak in the BOOMERanG data at l = 200

Characterization of Cosmic Microwave Background Temperature Fluctuations for Cosmological Topologies Specified by Topological Betti Numbers

The cosmic microwave background explorer, COBE, and the Balloon Observations of Millimetric Extragalactic Radiation and Geophysics, BOOMERanG, have collected data from the universe and detected relic anisotropies indicative of one of the earliest events of the universe, decoupling. Hidden in the correlation between these temperature fluctuations is the signature of the global shape, or topology, of the early universe. It is possible to calculate the temperature fluctuations as due to primeval adiabatic density temperature fluctuations from the Sachs-Wolfe effect, which contains a topological term. Here we investigate some of the spaces and how they affect the microwave background. A large class of spaces can be understood with some tools of topology associated with the way curves and volumes divide a space. The three tools of interest are the Euler Characteristic, the Betti numbers and fundamental domains. We intend to demonstrate the relationship between topology and anisotropy by creating a general equation that contains a topological term based on a topological invariant, the Betti Numbers of a manifold, and a term based on the temperature fluctuations. We will also extend the work of Silk on classifying the finite flat spaces through the use of a new three-dimensional plotting technique. These graphs demonstrate the inadequacy of the current CMB data\u27s topological predictive properties. As many shapes can produce the observed peaks. We will also extend the work of Inoue on classifying compact hyperbolic manifolds in two ways. First we demonstrate that the compact hyperbolic eigenmodes can be represented by a pFq function, and do not need to be calculated individually. Secondly, we extend the power spectrum plot the high angular resolution, large l, and compare compact hyperbolic manifolds to the observed peak in the BOOMERanG data at l = 200