In this thesis we classify primitive amalgams of degree (5,2) and thereby obtain a description of symmetric graphs of valency 5. There are 22 such amalgams, and together with this list, we find the presentations for the universal completions of such amalgams and an example of a faithful finite completion in each case. Our methods include group theory, the amalgam method and some modular represe...

;its minimal faithful permutation and matrix representation degrees are respectively 97 239 461 142 009 186 000 and 196 882. This makes it far harder to work with than the 25 smaller sporadics. In Section 3, we describe techniques which allow limited computation in the Monster. In Section 4, we describe our experiments to assess the effectiveness of combining these techniques with a suitable ve...

If M is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component XM of its SL(2,C)-character variety is an affine complex curve, which is smooth at the discrete faithful representation ρ0. Porti defined a non-abelian Reidemeister torsion in a neighborhood of ρ0 in XM and observed that it is an analytic map, which is the germ of a unique rational fun...

If M is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component XM of its SL(2,C)-character variety is an affine complex curve, which is smooth at the discrete faithful representation ρ0. Porti defined a non-abelian Reidemeister torsion in a neighborhood of ρ0 in XM and observed that it is an analytic map, which is the germ of a unique rational fun...

By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup S in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleannes present in the semilattice of idempotents of ...

We study random walks on the three-strand braid group B3, and in particular compute the drift, or average topological complexity of a random braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic Harper-Hofstadter problem), what enables to build a faithful representation of B3 as generalized magnetic t...

In the canonical approach to Lorentzian Quantum General Relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of a Hilbert space structure which was later proved to be a faithful representation of the canonical commutation and adjointness relations of the quantum field algebra of diffeomor...

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current st...

Hierarchical Markov networks (HMNs) were recently proposed as a faithful representation of Bayesian networks. In this paper, we propose a query processing algorithm for HMNs. This method takes one query processing algorithm for a traditional Markov network and extends it to a hierarchy of Markov networks. Experimental results explicitly demonstrate the effectiveness of our approach. The work he...

Given a finite dimensional Lie algebra g , let z ( ) denote the center of and μ be minimal possible dimension for faithful representation . In this paper we obtain L r 2 where k is free -step nilpotent rank particular prove that = ⌈ − 1 ⌉ + ≥ 4 It turns out ∼ dim ⁡ (as → ∞ present some evidence could true any This considerably lower than known bounds which are (for fixed polynomial in