# Réseaux de Coxeter-Davis et commensurateurs

Annales de l'institut Fourier (1998)

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

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topHaglund, 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

top- [1] W. BALLMAN, M. BRIN, Polygonal complexes and combinatorial group theory, Geom. Dedicata, 50, sér. I (1994), 165-191. Zbl0832.57002MR95e:57004
- [2] H. BASS, R. KULKARNI, Uniform tree lattices, J. Amer. Math. Soc., 3 (1990), 843-902. Zbl0734.05052MR91k:20034
- [3] N. BENAKLI, Polyèdres à géométrie locale donnée, C. R. Acad. Sci., Paris, sér. I 313, n° 9, 561-564. Zbl0744.57003MR92k:52018
- [4] M. BURGER, S. MOZES, CAT(-1) spaces, divergence groups and their commensurators, J. Amer. Math. Soc, 9 (1996), 57-94. Zbl0847.22004MR96c:20065
- [5] M. DAVIS, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math., 117 (1983), 293-324. Zbl0531.57041MR86d:57025
- [6] M. DAVIS, Buildings are CAT(0), prépublication, Ohio State University, 1994. Zbl0978.51005
- [7] Y. GAO, Superrigidity for homomorphisms into isometry groups of non proper CAT(-1) spaces, prépublication, Yale University, 1995. Zbl0888.22008
- [8] M. GROMOV, Infinite groups as geometric objects, dans Proceedings of the International Congress of Mathematicians, Varsovia, 1983.
- [9] A. HAEFLIGER, Complexes of groups and orbihedra, dans Group theory from a geometrical viewpoint, E. Ghys et A. Haefliger éd., World Scientific, 1991. Zbl0858.57013MR93m:20048
- [10] F. HAGLUND, Les polyèdres de Gromov, C. R. Acad. Sci., Paris, 313, sér. I, 603-606. Zbl0749.52011MR92m:57005
- [11] Y. LIU, Density of the commensurability group of uniform tree lattices, J. Alg., 165 (1994), 346-359. Zbl0813.20024MR95c:20036
- [12] G. MARGULIS, Discrete subgroups of semi-simple groups, Ergeb. Math. Grenz., 17, Springer Verlag, 1991. Zbl0732.22008MR92h:22021
- [13] G. MOUSSONG, Hyperbolic Coxeter groups, Thèse, Ohio State University, 1988.
- [14] F. PAULIN, Construction of hyperbolic groups via hyperbolization of polyhedra, dans Group theory from a geometrical viewpoint, E. Ghys et A. Haefliger éd., World Scientific, 1991.
- [15] M.A. RONAN, Lectures on buildings, Persp. Math., 7, Academic Press, 1989. Zbl0694.51001MR90j:20001
- [16] R. ZIMMER, Ergodic theory and semi-simple groups, Birkhauser, 1984. Zbl0571.58015MR86j:22014

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