The Euler characteristic of graph products and of Coxeter groups

1992 ◽  
pp. 36-46 ◽  
Author(s):  
I. M. Chiswell
Keyword(s):  
Euler Characteristic ◽  
Coxeter Groups ◽  
Graph Products

scholarly journals Bredon cohomological dimension for virtually abelian stabilisers for CAT(0) groups

2019 ◽  
pp. 1-13 ◽  
Author(s):  
Tomasz Prytuła
Keyword(s):  
Finite Groups ◽  
Discrete Group ◽  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Graph Products ◽  
The Family ◽  
Abelian Subgroups ◽  
Right Angled Artin Groups ◽  
Cube Complexes

Given a discrete group [Formula: see text], for any integer [Formula: see text] we consider the family of all virtually abelian subgroups of [Formula: see text] of rank at most [Formula: see text]. We give an upper bound for the Bredon cohomological dimension of [Formula: see text] for this family for a certain class of groups acting on CAT(0) spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of Lafont.

scholarly journals Euler Characteristics and imbeddings of hyperbolic Coxeter groups

1998 ◽  
Vol 64 (2) ◽  
pp. 149-161 ◽  
Author(s):  
George Maxwell
Keyword(s):  
Euler Characteristic ◽  
Hyperbolic Space ◽  
Coxeter Groups ◽  
Hyperbolic Groups ◽  
Fundamental Region ◽  
Euler Characteristics ◽  
Computer Aided ◽  
Level 1

AbstractThis paper has a twofold purpose. The first is to compute the Euler characteristics of hyperbolic Coxeter groups Ws of level 1 or 2 by a mixture of theoretical and computer aided methods. For groups of level 1 and odd values of |S|, the Euler characteristic is related to the volume of the fundamental region of Ws in hyperbolic space. Secondly we note two methods of imbedding such groups in each other. This reduces the amount of computation needed to determine the Euler characteristics and also reduces the number of essentially different hyperbolic groups that need to be considered.

scholarly journals SURFACE SUBGROUPS OF GRAPH PRODUCTS OF GROUPS

2012 ◽  
Vol 22 (08) ◽  
pp. 1240003
Author(s):  
SANG-HYUN KIM
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Coxeter Groups ◽  
Surface Group ◽  
Hyperbolic Surface ◽  
Graph Products ◽  
Graph Product ◽  
Cyclic Groups ◽  
Products Of Groups ◽  
Product Kernel

Let G be a graph product of a collection of groups and H be the direct product of the same collection of groups, so that there is a natural surjection p : G → H. The kernel of this map p is called a graph product kernel. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.

scholarly journals CoxIter – Computing invariants of hyperbolic Coxeter groups

2015 ◽  
Vol 18 (1) ◽  
pp. 754-773 ◽  
Author(s):  
R. Guglielmetti
Keyword(s):  
Computer Program ◽  
Euler Characteristic ◽  
Coxeter Groups ◽  
Source Code ◽  
Theoretical Background ◽  
Combinatorial Structure ◽  
Fundamental Polyhedron ◽  
Supplementary Material

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).Supplementary materials are available with this article.

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

scholarly journals Automorphisms of odd Coxeter groups

2021 ◽  
Author(s):  
Tushar Kanta Naik ◽  
Mahender Singh
Keyword(s):  
Coxeter Groups

scholarly journals The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Aleksander J. Cianciara ◽  
S. James Gates ◽  
Yangrui Hu ◽  
Renée Kirk
Keyword(s):  
Coxeter Groups ◽  
Auxiliary Field ◽  
Permutation Groups ◽  
Weight Space ◽  
Field Problem ◽  
Truncated Octahedron ◽  
Color Problem ◽  
Minimal Representations

Abstract A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

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