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Parameters of the two generator discrete elementary groups : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand
Let f, g be elements of M, the group of Möbius transformations of the extended complex plane Ĉ = C U ∞. We identify each element of M with a 2 × 2 complex matrix with determinant 1. The three complex numbers, β(f) = tr2
(f) - 4,β(g) = tr2
(g) - 4,γ(f,g) = tr[f,g] - 2, define the group ‹f,g› uniquely up to conjugacy whenever γ(f,g) ≠0; where tr(f) and tr(g) denote the traces of representive matrices of f and g respectively, [f,g] denotes the multiplicative commutator fgf-1
g-1
. We call these three complex numbers the parameters of ‹f,g›. This thesis is concerned with the parameters of discrete and elementary subgroups of M
Parameters of the two generator discrete elementary groups : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand
Let f, g be elements of M, the group of Möbius transformations of the extended complex plane Ĉ = C U ∞. We identify each element of M with a 2 × 2 complex matrix with determinant 1. The three complex numbers, β(f) = tr2
(f) - 4,β(g) = tr2
(g) - 4,γ(f,g) = tr[f,g] - 2, define the group ‹f,g› uniquely up to conjugacy whenever γ(f,g) ≠0; where tr(f) and tr(g) denote the traces of representive matrices of f and g respectively, [f,g] denotes the multiplicative commutator fgf-1
g-1
. We call these three complex numbers the parameters of ‹f,g›. This thesis is concerned with the parameters of discrete and elementary subgroups of M
Critical behavior in matrix model with branching interactions
Motivated by understanding the phase structure of strings we
investigate the matrix model with g' (\tr M(t)^{2})^{2} interaction
which is the simplest approximation of the model expected to describe the
critical phenomena of the large- reduced model of odd-dimensional matrix
field theory. We find three distinct phases: (i) an ordinary gravity
phase, (ii) a branched polymer phase and (iii) an intermediate phase. Further
we can also analyse the one with slightly generalized g^{(2)} (\frac{1}{N}\tr
M(t)^{2})^{2} +g^{(3)} (\frac{1}{N}\tr M(t)^{2})^{3} + \cdots + g^{(n)}
(\frac{1}{N}\tr M(t)^{2})^{n} interaction. As a result the multi-critical
versions of the phase (ii) are found.Comment: 11pages. latex (The arguments in Discussions are corrected and more
clarified! Several grammatical errors are corrected. And some preprints in
references are replaced with the published versions.
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