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Mod 3 arithmetic on triangulated Riemann surfaces

E.G. Belaga
2001 Theoretical Computer Science  
Let T be a triangulation of a Riemann surface, orientable or non-orientable and of an arbitrary genus. Suppose, a labelling of the vertices of T by three labels 0, +1, and −1 is ÿxed.  ...  paper deals with the following problem: ÿnd the number of labellings of the faces of T by two labels +1 and −1, in such a way that the sum of the labels of the faces around any vertex is equal modulo 3  ...  It means that the link between F 3 -arithmetic on a triangulation of a non-planar Riemann surface, on the one hand, and colourings of the vertices of this triangulation, on the other, ceases to exist (  ... 
doi:10.1016/s0304-3975(00)00236-x fatcat:bt75jhh2drap3ikgfic4rl56mq

Riemann–Roch like theorem for triangulated categories

Izuru Mori
2004 Journal of Pure and Applied Algebra  
We will prove a Riemann-Roch like theorem for triangulated categories satisfying Serre duality.  ...  As an application, we will prove Riemann-Roch Theorem and Adjunction Formula for noncommutative Cohen-Macaulay surfaces in terms of sheaf cohomology.  ...  In fact, one of the motivations of this paper is to apply Riemann-Roch Theorem and Adjunction Formula to a ÿber of a closed point and a K-theoretic section on a quantum ruled surface deÿned in [17] ,  ... 
doi:10.1016/j.jpaa.2004.03.008 fatcat:iopsocnuangpriqhehvnuik7oi

From the Borel–Serre compactification to curve complex of surfaces

Lizhen Ji
2018 Notices of the International Congress of Chinese Mathematicians  
In this paper, we describe the interaction and similarity between locally symmetric spaces and moduli spaces of Riemann surfaces, through the example of how the Borel-Serre compactification of locally  ...  symmetric spaces led to the curve complex of surfaces, which is a fundamental object in low dimensional topology and geometric group theory.  ...  The group Mod g acts on T g by changing the markings of Riemann surfaces, and we have the following identification: M g = Mod g \T g . When g = 1, T 1 = H 2 , and Mod 1 = SL(2, Z).  ... 
doi:10.4310/iccm.2018.v6.n2.a5 fatcat:g57fwydbanao3lkkbl2tsesgau

GPU-based visualization of domain-coloured algebraic Riemann surfaces [article]

Stefan Kranich
2015 arXiv   pre-print
The approach faithfully reproduces the topology and the holomorphic structure of the Riemann surface.  ...  We examine an algorithm for the visualization of domain-coloured Riemann surfaces of plane algebraic curves.  ...  For example, the point with index k is part of the triangle whose vertices have indices k − (k mod 3), k − (k mod 3) + 1, and k − (k mod 3) + 2.  ... 
arXiv:1507.04571v3 fatcat:yyrvyboj2ffpjnj3pdcsapjw3e

A panaroma of the fundamental group of the modular orbifold [article]

A. Muhammed Uludağ, Ayberk Zeytin
2016 arXiv   pre-print
We also discuss arithmetic related questions which exist in both the tame part (via Belyi's theorem) and the non-tame part.  ...  We are thankful to Athanase Papadopoulos for inviting us to publish in this volume and for his comments on the previous versions of this manuscript.  ...  That is to say, punctures on the Riemann surface correspond to finite left turn loops on the graph, infinite Farey branches correspond to boundaries of the Riemann surface that are homotopic to the circle  ... 
arXiv:1602.00283v1 fatcat:rfzfdbgfr5d4nndj76r3ilvtci

Page 3812 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
(F-STRAS; Strasbourg) Mod 3 arithmetic on triangulated Riemann surfaces. (English summary) Combinatorics and computer science (Palaiseau, 1997). Theoret. Comput. Sci. 263 (2001), no. 1-2, 123-137.  ...  Let T be a triangulation of a Riemann surface of arbitrary genus, and consider a fixed (0, 1, —1)-labelling of the vertices of 7.  ... 

Page 953 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
Modular invariance for su(3) theories; 3. Fermat curves; 4. Combinatorial groups for triangulated surfaces; 5. The Riemann surface of a RCFT on the torus; 6. Conclusions: A.  ...  In an attempt to go beyond the su(3) case, they show that any rational conformal field theory determines canonically a Riemann surface. This article is a detailed survey on these subjects.  ... 

Curve Complexes Versus Tits Buildings: Structures and Applications [chapter]

Lizhen Ji
2011 Springer Proceedings in Mathematics  
class groups of surfaces and the geometry and topology of 3-dimensional manifolds.  ...  subgroups and S-arithmetic subgroups of G(Q).  ...  This result was used to prove the quasi-isometric rigidity of Mod g,n in [6] [65] . See §7.4.  ... 
doi:10.1007/978-1-4614-0709-6_6 fatcat:5lwlglxepjgivan3hb3z76e5nm

Well-rounded equivariant deformation retracts of Teichmüller spaces [article]

Lizhen Ji
2014 arXiv   pre-print
., _g-equivariant deformation retracts, of the Teichmüller space _g of compact Riemann surfaces of genus g.  ...  We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.  ...  a flow on the unit disk that explains a problem with the spine in [Th].  ... 
arXiv:1302.0877v2 fatcat:g3u5hvh5pnf35ocpnaepbjkwju

Computational topology and normal surfaces: Theoretical and experimental complexity bounds [chapter]

Benjamin A. Burton, João Paixão, Jonathan Spreer
2013 2013 Proceedings of the Fifteenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
Experimentally, we study the worst-case and average-case complexities over a comprehensive body of roughly three billion input triangulations.  ...  In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance.  ...  There is a family C n of closed 1-vertex triangulations with n tetrahedra, n ≥ 5, each containing a vertex normal surface with maximum coordinate at least F n−3 if n ≡ 2 mod 3 or at least 2 F n−3 otherwise  ... 
doi:10.1137/1.9781611972931.7 dblp:conf/alenex/BurtonPS13 fatcat:3gryn6vayzfchbliu4u7l6zb7e

The virtual cohomology dimension of Teichmüller modular groups: the first results and a road not taken [article]

Nikolai V. Ivanov
2015 arXiv   pre-print
The paper is devoted to a detailed exposition of results outlined in author's 1983 note "On the virtual cohomology dimension of the Teichm\"uller modular group".  ...  Recall that Mod g naturally acts on the Teichmüller space T g of marked Riemann surfaces of genus g.  ...  The proof of the inequality vcd(Mod g ) 6g 7 is much more deep and is based on theories of Riemann surfaces and of Teichmüller spaces.  ... 
arXiv:1510.00956v1 fatcat:4nosu2uifjdf7ehzuyqpxjerlm

Author index volume 263 (2001)

2001 Theoretical Computer Science  
., Mod 3 arithmetic on triangulated Riemann surfaces (1}2) 123}137 CaleH gari, P., F. Guidec, P. Kuonen and F.  ...  CleH menc7 on, Voronoi diagrams on piecewise #at surfaces and an application to biological growth (1}2) 263}274 KaH rolyi, G. and V.  ... 
doi:10.1016/s0304-3975(01)00283-3 fatcat:xv3da2ixwzf4pfp47jgune5zg4

The quasi-additivity law in conformal geometry

Jeremy Kahn, Mikhail Lyubich
2009 Annals of Mathematics  
On a Riemann surface S of finite type containing a family of N disjoint disks D i ("islands"), we consider several natural conformal invariants measuring the distance from the islands to ∂S and the separation  ...  Then mod(U, A) ≤ mod(V, B) ≤ N mod(U, A). Lemma 4 . 6 . 46 Let f : U → V be a branched covering between two compact Riemann surfaces with boundary.  ...  By an "arithmetic inequality" of Section 2.8, this yields the desired quadratic relation between the moduli X, Y and Z. 2.2. Paths and rectangles. Let S be a Riemann surface with boundary.  ... 
doi:10.4007/annals.2009.169.561 fatcat:37tgauwaavayjkdmqlaep2k7g4

Equivelar maps on the torus

Ulrich Brehm, Wolfgang Kühnel
2008 European journal of combinatorics (Print)  
These are precisely the ones which are quotients of the regular tessellations {3, 6}, {6, 3} or {4, 4} by a pure translation group.  ...  maps or type {3, 6} (or {4, 4}, resp.).  ...  An enumeration of the huge number of rooted maps on the torus in a very general sense can be found in [3] . For a survey about polyhedral maps on surfaces see [6] . Definition.  ... 
doi:10.1016/j.ejc.2008.01.010 fatcat:eiiahlbf5nbplkzcxzz75wdwli

Comments on the links between su(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards

M. Bauer, A. Coste, C. Itzykson, P. Ruelle
1997 Journal of Geometry and Physics  
In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface.  ...  We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves.  ...  Riemann surfaces.  ... 
doi:10.1016/s0393-0440(96)00027-7 fatcat:fcxmk2emdncpjeaqp5m5fn4p5i
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