Réseaux de Coxeter-Davis et commensurateurs

Frédéric Haglund

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 3, page 649-666
  • ISSN: 0373-0956

Abstract

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For each integer k ≥ 6 and each finite graph L , we construct a Coxeter group W and a non positively curved polygonal complex A on which W acts properly cocompactly, such that each polygon of A has k edges, and the link of each vertex of A is isomorphic to L . If L is a “generalized m -gon”, then A is a Tits building modelled on a reflection group of the hyperbolic plane. We give a condition on Aut ( L ) for Aut ( A ) to be non enumerable (which is satisfied if L is a thick classical generalized m -gon). On the other hand, if L has no loops of length 3 and if 4 divides k , we prove an arithmeticity property for W : the commensurator of W in Aut ( A ) is dense in Aut ( A ) .

How to cite

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Haglund, Frédéric. "Réseaux de Coxeter-Davis et commensurateurs." Annales de l'institut Fourier 48.3 (1998): 649-666. <http://eudml.org/doc/75297>.

@article{Haglund1998,
abstract = {For each integer $k\ge 6$ and each finite graph $L$, we construct a Coxeter group $W$ and a non positively curved polygonal complex $A$ on which $W$ acts properly cocompactly, such that each polygon of $A$ has $k$ edges, and the link of each vertex of $A$ is isomorphic to $L$. If $L$ is a “generalized $m$-gon”, then $A$ is a Tits building modelled on a reflection group of the hyperbolic plane. We give a condition on $\{\rm Aut\}(L)$ for $\{\rm Aut\}(A)$ to be non enumerable (which is satisfied if $L$ is a thick classical generalized $m$-gon). On the other hand, if $L$ has no loops of length 3 and if 4 divides $k$, we prove an arithmeticity property for $W$: the commensurator of $W$ in $\{\rm Aut\}(A)$ is dense in $\{\rm Aut\}(A)$.},
author = {Haglund, Frédéric},
journal = {Annales de l'institut Fourier},
keywords = {polygonal complexes; CAT(0) spaces; Coxeter groups; commensurators; Tits buildings},
language = {eng},
number = {3},
pages = {649-666},
publisher = {Association des Annales de l'Institut Fourier},
title = {Réseaux de Coxeter-Davis et commensurateurs},
url = {http://eudml.org/doc/75297},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Haglund, Frédéric
TI - Réseaux de Coxeter-Davis et commensurateurs
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 3
SP - 649
EP - 666
AB - For each integer $k\ge 6$ and each finite graph $L$, we construct a Coxeter group $W$ and a non positively curved polygonal complex $A$ on which $W$ acts properly cocompactly, such that each polygon of $A$ has $k$ edges, and the link of each vertex of $A$ is isomorphic to $L$. If $L$ is a “generalized $m$-gon”, then $A$ is a Tits building modelled on a reflection group of the hyperbolic plane. We give a condition on ${\rm Aut}(L)$ for ${\rm Aut}(A)$ to be non enumerable (which is satisfied if $L$ is a thick classical generalized $m$-gon). On the other hand, if $L$ has no loops of length 3 and if 4 divides $k$, we prove an arithmeticity property for $W$: the commensurator of $W$ in ${\rm Aut}(A)$ is dense in ${\rm Aut}(A)$.
LA - eng
KW - polygonal complexes; CAT(0) spaces; Coxeter groups; commensurators; Tits buildings
UR - http://eudml.org/doc/75297
ER -

References

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