scholarly journals Coxeter group in Hilbert geometry

2017 ◽  
Vol 11 (3) ◽  
pp. 819-877 ◽  
Author(s):  
Ludovic Marquis
Keyword(s):  
Coxeter Group ◽  
Hilbert Geometry

Tits Indices

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Coxeter Group ◽  
Coxeter System ◽  
Relative Rank ◽  
Coxeter Diagram ◽  
The Absolute ◽  
Relative Type

This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.

scholarly journals Bounding reflection length in an affine Coxeter group

2011 ◽  
Vol 34 (4) ◽  
pp. 711-719 ◽  
Author(s):  
Jon McCammond ◽  
T. Kyle Petersen
Keyword(s):  
Coxeter Group

scholarly journals The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

2014 ◽  
Vol 66 (2) ◽  
pp. 354-372 ◽  
Author(s):  
Ruth Kellerhals ◽  
Alexander Kolpakov
Keyword(s):  
Growth Rate ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Triangle Group ◽  
Minimal Growth ◽  
Salem Number

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

scholarly journals Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

2011 ◽  
Vol 16 ◽  
pp. 82 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Muna Al-Shueili
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Linear Combinations ◽  
Octahedral Group ◽  
Snub Cube ◽  
Proper Rotation

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand  respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.  

scholarly journals From Funk to Hilbert geometry

2014 ◽  
pp. 33-67
Author(s):  
Athanase Papadopoulos ◽  
Marc Troyanov
Keyword(s):  
Hilbert Geometry

scholarly journals Quotient Isomorphism Invariants of a Finitely Generated Coxeter Group

2008 ◽  
Author(s):  
Michael Mihalik ◽  
John Ratcliffe ◽  
Steven Tschantzk
Keyword(s):  
Coxeter Group ◽  
Finitely Generated
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