Search

A two-generator Kleinian group $\langle f,g \rangle$ can be naturally associated with a discrete group $\langle f,\phi \rangle$ with the generator $\phi$ of order $2$ and where \begin{equation*} \langle f,\phi f \phi^{-1} \rangle= \langle f,gfg^{-1} \rangle \subset \langle f,g\rangle, \quad [ \langle f,g f g^{-1} \rangle: \langle f,\phi \rangle]=2 \end{equation*} This is useful in studying the geometry of Kleinian groups since $\langle f,g \rangle$ will be discrete only if $\langle f,\phi \rangle$ is, and the moduli space of groups $\langle f,\phi \rangle$ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of $\langle f,g \rangle$. The dimension reduction here is realised by a projection of principal characters of two-generator Kleinian groups. In applications it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from J\o rgensen's results on algebraic convergence.

As a result of the Hund's coupling, the band structure of the conducting electrons in the skyrmion crystal (SkX) shares similar topological properties with that of graphene, such as its cone-like shape, nonzero band Chern number, edge states, and etc. In this work, we rigorously demonstrate that the Klein tunneling phenomena is also shared by these two. We use the Green's function technique and calculated the transmission probability of the electrons tunneling through an electrostatic barrier in the SkX expressed by the double exchange model. Numerical results of the SkX reproduced the Dirac model obtained by linear fitting the two-dimensional band structure of the SkX.

The principal character of a representation of the free group of rank two into PSL(2, C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group that is not virtually abelian, that is a Kleinian group. A classical necessary condition is J{\o}rgensen's inequality. Here we use certainly shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself., Comment: 18 pages

By considering the bulk-edge correspondence of the skyrmion crystal this work focused on the relation of the topological properties to the polarity ($Q_{\rm{sk}}$), next-nearest-neighbor hopping ($t'$), and exchange coupling ($J$). We found that by continually increasing polarity of the skyrmions from 1 to 2, the monopole-lattice phase and the dipole-lattice phase both span a wide range in the parameter space demonstrating topological robustness. While the next-nearest-neighbor-hopping of the electrons coexists with the nearest-neighbor hopping the flux pattern of the emergent magnetic field traversing the two-dimensional SkX is reshaped giving rise to different topological Hall effect. By increasing $t'$ from 0 to $t$ with the polarity fixed to be 1, we found Chern numbers of the electronic bands change from unity to indefinite values demonstrating a phase transition. We also considered the effect of finite $J$ and found the results obtained in the adiabatic limit hold to around $J=t$.

This paper considers generalizations of open mappings, closed mappings, pseudo-open mappings, and quotient mappings from topological spaces to generalized topological spaces. Characterizations of these classes of mappings are obtained and some relationships among these classes are established., Comment: 8 papges