Near integral domains on nonabelian groups

In this paper the concept of a special automorphism is introduced and used to analyze near integral domains having nonabelian addtive groups. We show that there are finite and infinite near integral domains having additive groups with arbitrary class of nilpotency. We also give another example of a non-nilpotent group which is the additive group of a near integral domain. Finally, nonabelian groups of order less than 1000 are examined to determine which can be the additive group of a near integral domain.Most of the results of this paper are contained in the author's doctoral dissertation at Boston University. The author thanks ProfessorD. W. Blackett for his guidance.     

Turing degrees of nonabelian groups

For a countable structure , the (Turing) degree spectrum of is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of has the least degree , then we say that is the (Turing) degree of the isomorphism type of . So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.

     

Bohr compactifications of nonabelian Lie semigroups

Communicated by K. H. Hofmann     

On Calabi-Yau threefolds with large nonabelian fundamental groups

In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order as quotients of the small resolutions of certain complete intersections of quadrics in that were first considered by M. Gross and S. Popescu.

     

Groups with maximum condition for nonabelian subgroups

We study locally soluble-by-finite groups with the maximum condition for nonabelian subgroups. These groups do not necessarily satisfy the maximum condition for all subgroups. But they are finitely generated and metabelian-byfinite.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 925–930, July–August, 1991.     

Witten's nonabelian localization for noncompact Hamiltonian spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303-368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.     

Tight frames generated by finite nonabelian groups

Let $\cal H$ be a Hilbert space of finite dimension d, such as the finite signals ? ₂(d) or a space of multivariate orthogonal polynomials, and n?≥?d. There is a finite number of tight frames of n vectors for $\cal H$ which can be obtained as the orbit of a single vector under the unitary action of an abelian group G (of symmetries of the frame). Each of these so called harmonic frames or geometrically uniform frames can be obtained from the character table of G in a simple way. These frames are used in signal processing and information theory. For a nonabelian group G there are in general uncountably many inequivalent tight frames of n vectors for $\cal H$ which can be obtained as such a G-orbit. However, by adding an additional natural symmetry condition (which automatically holds if G is abelian), we obtain a finite class of such frames which can be constructed from the character table of G in a similar fashion to the harmonic frames. This is done by identifying each G-orbit with an element of the group algebra ?G (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.     

Malliavin's theorem for weak synthesis on nonabelian groups

Malliavin's celebrated theorem on the failure of spectral synthesis for the Fourier algebra A(G) on nondiscrete abelian groups was strengthened to give failure of weak synthesis by Parthasarathy and Varma. We extend this to nonabelian groups by proving that weak synthesis holds for A(G) if and only if G is discrete. We give the injection theorem and the inverse projection theorem for weak X-spectral synthesis, as well as a condition for the union of two weak X-spectral sets to be weak X-spectral for an A(G)-submodule X of VN(G). Relations between weak X-synthesis in A(G) and A(G×G) and the Varopoulos algebra V(G) are explored. The concept of operator synthesis was introduced by Arveson. We extend several recent investigations on operator synthesis by defining and studying, for a V^∞(G)-submodule M of B(L²(G)), sets of weak M-operator synthesis. Relations between X-Ditkin sets and M-operator Ditkin sets and between weak X-spectral synthesis and weak M-operator synthesis are explored.     

A structure theorem for small sumsets in nonabelian groups

Let G$G$ be an arbitrary finite group and let S$S$ and T$T$ be two subsets such that |S|≥2$\left|S\right|\ge 2$, |T|≥2$\left|T\right|\ge 2$, and |TS|≤|T|+|S|−1≤|G|−2$\left|TS\right|\le \left|T\right|+\left|S\right|-1\le \left|G\right|-2$. We show that if |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ then either S$S$ is a geometric progression or there exists a non-trivial subgroup H$H$ such that either |HS|≤|S|+|H|−1$\left|HS\right|\le \left|S\right|+\left|H\right|-1$ or |SH|≤|S|+|H|−1$\left|SH\right|\le \left|S\right|+\left|H\right|-1$. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ we show the existence of counterexamples to the above characterization whose structure is described precisely.     

Twisted Alexander polynomials of twist knots for nonabelian representations

In this paper, we describe the twisted Alexander polynomial of twist knots for nonabelian SL(2,C)-representations and investigate in detail the coefficient of the highest degree term as a function on the representation space of the knot group. In particular, we introduce the notion of monic representation and discuss its relation to the fiberedness of knots.