scholarly journals Two-dimensional Veronese groups with an invariant ball

2017 ◽  
Vol 28 (10) ◽  
Author(s):  
Angel Cano ◽  
Luis Loeza
Keyword(s):  
Surface Group ◽  
Compact Surface ◽  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
Two Dimensional

In this paper, we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in [Formula: see text]. As a corollary we get that every discrete compact surface group in [Formula: see text] admits a deformation in [Formula: see text] with a nonempty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on [Formula: see text].

One-relator surface groups

1990 ◽  
Vol 108 (3) ◽  
pp. 467-474 ◽  
Author(s):  
John Hempel
Keyword(s):  
Normal Subgroup ◽  
Single Point ◽  
Surface Group ◽  
Compact Surface ◽  
Heegaard Splitting ◽  
Normal Closure ◽  
Single Element ◽  
Homotopy Classes ◽  
Simple Loop ◽  
Proper Power

For X a subset of a group G, the smallest normal subgroup of G which contains X is called the normal closure of X and is denoted by ngp (X; G) or simply by ngp (X) if there is no possibility of ambiguity. By a surface group we mean the fundamental group of a compact surface. We are interested in determining when a normal subgroup of a surface group contains a simple loop – the homotopy class of an embedding of S1 in the surface, or more generally, a power of a simple loop. This is significant to the study of 3-manifolds since a Heegaard splitting of a 3-manifold is reducible (cf. [2]) if and only if the kernel of the corresponding splitting homomorphism contains a simple loop. We give an answer in the case that the normal subgroup is the normal closure ngp (α) of a single element α: if ngp (α) contains a (power of a) simple loop β then α is homotopic to a (power of a) simple loop and β±1 is homotopic either to (a power of) α or to the commutator [α, γ] of a with some simple loop γ meeting a transversely in a single point. This implies that if a is not homotopic to a power of a simple loop, then the quotient map π1(S) → π1(S)/ngp (α) does not factor through a group with more than one end. In the process we show that π1(S)/ngp (α) is locally indicable if and only if α is not a proper power and that α always lifts to a simple loop in the covering space Sα of S corresponding to ngp (α). We also obtain some estimates on the minimal number of double points in certain homotopy classes of loops.

The effects of surface group functionalization and strain on the electronic structures of two-dimensional silicon carbide

2015 ◽  
Vol 628 ◽  
pp. 60-65
Author(s):  
Peng Zhang ◽  
Xiuli Hou ◽  
Yanqiong He ◽  
Qiuming Peng ◽  
Mingdong Dong
Keyword(s):  
Silicon Carbide ◽  
Electronic Structures ◽  
Surface Group ◽  
Two Dimensional

Design of a Three-Axis Surface Encoder with a Blue-Ray Laser Diode

2012 ◽  
Vol 523-524 ◽  
pp. 913-918 ◽  
Author(s):  
Xing Hui Li ◽  
Yuki Shimizu ◽  
Hiroshi Muto ◽  
So Ito ◽  
Wei Gao
Keyword(s):  
Laser Beam ◽  
Laser Diode ◽  
Short Wavelength ◽  
Path Difference ◽  
Compact Surface ◽  
Two Dimensional ◽  
Light Path ◽  
Basic Measurement ◽  
Interference Signals ◽  
Measurement Principle

A compact optical three-axis surface encoder, which can detect displacements in the XYZ-directions simultaneously by employing a scale grating with a short pitch (0.57 μm) and a single laser beam with a short wavelength (405 nm), is described in this paper. This surface encoder mainly consists of a blue-ray laser diode, a pair of two-dimensional diffractive gratings serving as scale grating and reference grating, respectively, and two quarter photodiodes (QPD) utilized for detecting output optical density. The displacements of the scale grating along X-, Y- and Z-directions are measured by analyzing interference signals caused by the phase shifts and light path difference of diffracted beams from the scale grating and reference grating. Basic measurement principle is illuminated, a compact surface encoder is designed and evaluation experiments are carried out.

scholarly journals Quasiconvexity and Amalgams

1997 ◽  
Vol 07 (06) ◽  
pp. 771-811 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Cyclic Subgroup ◽  
Hyperbolic Group ◽  
Hyperbolic Surface ◽  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
3 Dimensional ◽  
Word Hyperbolic Group ◽  
Amalgamated Free Product ◽  
One Relator Groups ◽  
Word Hyperbolic Groups

We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian groups have property (Q). We also give some applications of our results to one-relator groups and exponential groups.

scholarly journals A two-dimensional slice through the parameter space of two-generator Kleinian groups

2018 ◽  
Vol 28 (08) ◽  
pp. 1535-1564
Author(s):  
Elena Klimenko ◽  
Natalia Kopteva
Keyword(s):  
Parameter Space ◽  
Kleinian Groups ◽  
Two Dimensional ◽  
Real Trace ◽  
Discreteness Criteria

We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this, we gather together known discreteness criteria for two-generator groups with a parabolic generator and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for groups generated by a parabolic and a [Formula: see text]-loxodromic elements whose commutator has real trace and present all orbifolds uniformized by such groups.

scholarly journals Unification and classification of two-dimensional crystalline patterns using orbifolds

2014 ◽  
Vol 70 (4) ◽  
pp. 319-337 ◽  
Author(s):  
S. T. Hyde ◽  
S. J. Ramsden ◽  
V. Robins
Keyword(s):  
Euclidean Space ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Groups ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
Simple Rules

The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway's orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler `crystallographic hyperbolic groups' are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (i.e.theP,D,Gyroid,CLPandHsurfaces) as well as the genus-fourI-WPsurface.

scholarly journals Boundary Convex Cocompactness and Stability of Subgroups of Finitely Generated Groups

2017 ◽  
Vol 2019 (6) ◽  
pp. 1699-1724
Author(s):  
Matthew Cordes ◽  
Matthew Gentry Durham
Keyword(s):  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Limit Set ◽  
Class Groups ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Equivalent Characterization ◽  
Convex Cocompact

Abstract A Kleinian group $\Gamma < \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $\Gamma$ in $\mathbb H^3$ is quasiconvex or, equivalently, $\Gamma$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.

scholarly journals A Numerical Lower Bound for the Spectral Radius of Random Walks on Surface Groups

2015 ◽  
Vol 24 (6) ◽  
pp. 838-856 ◽  
Author(s):  
S. GOUEZEL
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Spectral Radius ◽  
Cayley Graphs ◽  
Surface Group ◽  
Hyperbolic Groups ◽  
Surface Groups ◽  
Genus 2 ◽  
Order Of Magnitude

Estimating numerically the spectral radius of a random walk on a non-amenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

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