scholarly journals Affine Equivalence and Saddle Connection Graphs of Half-Translation Surfaces

2020 ◽  
Author(s):  
Huiping Pan
Keyword(s):  
Automorphism Group ◽  
Translation Surface ◽  
Saddle Connection ◽  
Translation Surfaces ◽  
Affine Equivalence ◽  
Quotient Graph ◽  
Affine Homeomorphism ◽  
Connection Graph

Abstract To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the saddle connection graph and the corresponding quotient graph.

scholarly journals Graph polynomials and symmetries

2019 ◽  
Vol 18 (09) ◽  
pp. 1950172 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Prime Order ◽  
Tutte Polynomial ◽  
Quotient Graph ◽  
Graph Polynomials ◽  
Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

scholarly journals Some New Characterizations Of Parallel Translation Surface According To Bishop Frame With Timelike M1 In Minkowski 3-Space

2018 ◽  
Vol 22 ◽  
pp. 01030
Author(s):  
Gülden Altay Suroǧlu
Keyword(s):  
Translation Surface ◽  
Parallel Translation ◽  
Translation Surfaces ◽  
Bishop Frame

In this paper we consider parallel translation surfaces, which are generated by spacelike curves, according to Bishop frame with timelike M1 in Minkowski 3- space. Then, we obtain some characterizations of these surface.

CONNECTIVITY AND SPECTRUM IN A GRAPH WITH A REGULAR AUTOMORPHISM GROUP OF ODD ORDER

1994 ◽  
Vol 04 (04) ◽  
pp. 529-560 ◽  
Author(s):  
JON A. SJOGREN
Keyword(s):  
Automorphism Group ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Regular Automorphism ◽  
Fixed Integer ◽  
Quotient Graph ◽  
Odd Order ◽  
Induced Graph ◽  
Group Matrices

Let a finite group G of odd order n act regularly on a connected (multi-)graph Γ. That is, no group element other than the identity fixes any vertex. Then the “quotient graph” Δ under the action is the induced graph of orbits. We give a result about the connectivity of Γ and Δ in terms of their numbers of labeled spanning trees. In words, the spanning tree count of the graph is equal to n, the order of the given regular automorphism group, times the spanning tree count of the graph of orbits, times a perfect square integer. There is a dual result on the Laplacian spectrum saying that the multiset of Laplacian eigenvalues for the main graph is the disjoint union of the multiset for the quotient graph together with a multiset all of whose elements have even multiplicity. Specializing to the case of one orbit, we observe that a Cayley graph of odd order has spanning tree count equal to n times a square, and that that the Laplacian spectrum consists of the value 0 together with other doubled eigenvalues. These results are based on a study of matrices (and determinants) that consist of blocks of group-matrices. The generic determinant for such a matrix with the additional property of symmetry will have a dominanting square factor in its (multinomial) factorization. To show this, we make use of the Feit-Thompson theorem which provides a normal tower for an odd-order group, and perform a similarity conjugation with a fixed integer, unimodal matrix. Additional related results are given for certain group-matrices “twisted” by a group of automorphisms, generalizing the “g-circulants” of P.J. Davis.

scholarly journals Gaps of saddle connection directions for some branched covers of tori

2021 ◽  
pp. 1-55
Author(s):  
ANTHONY SANCHEZ
Keyword(s):  
Moduli Space ◽  
Saddle Connection ◽  
Branched Covers ◽  
Translation Surfaces ◽  
Horocycle Flow ◽  
Return Times ◽  
Marked Points ◽  
Specific Orientation

Abstract We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at zero and quadratic tail decay. We also construct examples of translation surfaces in any genus $d>1$ that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked tori and saddle connections between distinct marked points with a specific orientation. These results can be interpreted as the gap distribution of slopes of affine lattices. We obtain our results by translating the question of gap distributions to a dynamical question of return times to a transversal under the horocycle flow on an appropriate moduli space.

scholarly journals Weighted minimal translation surfaces in the Galilean space with density

2017 ◽  
Vol 15 (1) ◽  
pp. 459-466 ◽  
Author(s):  
Dae Won Yoon
Keyword(s):  
Mean Curvature ◽  
Linear Density ◽  
Translation Surface ◽  
Plane Curves ◽  
Isotropic Plane ◽  
Translation Surfaces ◽  
Weighted Mean ◽  
Weighted Mean Curvature ◽  
Galilean Space ◽  
Log Linear

Abstract Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

scholarly journals Finite blocking property versus pure periodicity

2009 ◽  
Vol 29 (3) ◽  
pp. 983-996 ◽  
Author(s):  
THIERRY MONTEIL
Keyword(s):  
Finite Number ◽  
Periodic Trajectory ◽  
Translation Surface ◽  
Translation Surfaces ◽  
Branched Coverings ◽  
Directional Flow ◽  
Blocking Property ◽  
Genus Two ◽  
Property Versus ◽  
Null Measure

AbstractA translation surface 𝒮 is said to have the finite blocking property if for every pair (O,A) of points in 𝒮 there exists a finite number of ‘blocking’ points B1,…,Bn such that every geodesic from O to A meets one of the Bis. 𝒮 is said to be purely periodic if the directional flow is periodic in each direction whose directional flow contains a periodic trajectory (this implies that 𝒮 admits a cylinder decomposition in such directions). We will prove that the finite blocking property implies pure periodicity. We will also classify the surfaces that have the finite blocking property in genus two: such surfaces are exactly the torus branched coverings. Moreover, we prove that in every stratum such surfaces form a set of null measure. In Appendix A, we prove that completely periodic translation surfaces form a set of null measure in every stratum.

scholarly journals Large-scale geometry of the saddle connection graph

2021 ◽  
pp. 1
Author(s):  
Valentina Disarlo ◽  
Huiping Pan ◽  
Anja Randecker ◽  
Robert Tang
Keyword(s):  
Large Scale ◽  
Saddle Connection ◽  
Large Scale Geometry ◽  
Connection Graph

scholarly journals Classification of j-maximal spacelike affine translation surfaces in the Minkovski space i31 with density

2019 ◽  
Vol 15 (3) ◽  
pp. 36
Author(s):  
Tran Le Nam
Keyword(s):  
Minkowski Space ◽  
Translation Surface ◽  
Maximal Surface ◽  
Translation Surfaces ◽  
Bernstein Theorem ◽  
Lagrange’S Equation ◽  
Affine Translation ◽  
Two Parameters

An affine translation surface is a graph of a function   introduced by Liu and Yu in 2013. The article considers the spacelike affine translation surfaces in the Minkowski space  with density  establishing the Lagrange’s equation type for -maximal surface, classifying -maximal spacelike affine translation surfaces. The result obtains two parameters and . From that, the Calabi – Bernstein theorem in this space is not true because two function  and  are defined on  

scholarly journals Dual translation surfaces in the three dimensional simply isotropic space $\mathbb{I}_{3}^{1}$

2018 ◽  
Vol 49 (1) ◽  
pp. 67-77
Author(s):  
Mohamd Saleem Lone ◽  
Murat Kemal Karacan
Keyword(s):  
Mean Curvature ◽  
Three Dimensional ◽  
Translation Surface ◽  
Translation Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
Dual Translation

In this paper, we study the dual translation surfaces in three dimensional simply isotropic space. We give classification of dual translation surface with constant dual isotropic mean curvature or constant dual isotropic Guassian curvature.

scholarly journals Ergodicity for infinite periodic translation surfaces

2013 ◽  
Vol 149 (8) ◽  
pp. 1364-1380 ◽  
Author(s):  
Pascal Hubert ◽  
Barak Weiss
Keyword(s):  
Translation Surface ◽  
Translation Surfaces ◽  
Lattice Surface

AbstractFor a $ \mathbb{Z} $-cover $\widetilde {M} \rightarrow M$ of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.

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