location:  Publications → journals
Search results

Search: All articles in the CJM digital archive with keyword Coxeter group

 Expand all        Collapse all Results 1 - 4 of 4

1. CJM 2013 (vol 66 pp. 323)

Hohlweg, Christophe; Labbé, Jean-Philippe; Ripoll, Vivien
 Asymptotical behaviour of roots of infinite Coxeter groups Let $W$ be an infinite Coxeter group. We initiate the study of the set $E$ of limit points of normalized'' roots (representing the directions of the roots) of W. We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form $B$ associated to a geometric representation, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, and then we exhibit a countable subset of $E$, formed by limit points for the dihedral reflection subgroups of $W$. We explain how this subset is built from the intersection with $Q$ of the lines passing through two positive roots, and finally we establish that it is dense in $E$. Keywords:Coxeter group, root system, roots, limit point, accumulation setCategories:17B22, 20F55

2. CJM 2013 (vol 66 pp. 354)

Kellerhals, Ruth; Kolpakov, Alexander
 The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has 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 ${\mathbb H^2}$ 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). Keywords:hyperbolic Coxeter group, growth rate, Salem numberCategories:20F55, 22E40, 51F15

3. CJM 2009 (vol 61 pp. 740)

Caprace, Pierre-Emmanuel; Haglund, Frédéric
 On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if $\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0) realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings. Keywords:Coxeter group, flat rank, $\cat0$ space, buildingCategories:20F55, 51F15, 53C23, 20E42, 51E24

4. CJM 2005 (vol 57 pp. 844)

Williams, Gordon
 Petrie Schemes Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Gr\"unbaum--Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. Keywords:Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter groupCategories:52B15, 52B05
 top of page | contact us | privacy | site map |