Determination up to isomorphism of right-angled Coxeter systems. (English) Zbl 1054.20021
Summary: We announce that every right-angled Coxeter group determines its Coxeter system up to isomorphism. This implies that Dranishnikov’s rigidity conjecture holds for right-angled Coxeter groups, i.e., every right-angled Coxeter group determines its boundary up to homeomorphism.
MSC:
20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
20F65 | Geometric group theory |
57M07 | Topological methods in group theory |
Keywords:
right-angled Coxeter groups; Coxeter systems; Dranishnikov rigidity conjecture; boundaries of groupsReferences:
[1] | Bourbaki, N.: Groupes et Algebrès de Lie. Chapters IV-VI, Masson, Paris (1981). · Zbl 0483.22001 |
[2] | Bridson, M. R., and Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer-Verlag, Berlin (1999). · Zbl 0988.53001 |
[3] | Brown, K. S.: Buildings. Springer-Verlag, Berlin (1980). |
[4] | Charney, R., and Davis, M. W.: When is a Coxeter system determined by its Coxeter group? J. London Math. Soc., 61 (2), 441-461 (2000). · Zbl 0983.20034 · doi:10.1112/S0024610799008583 |
[5] | Davis, M. W.: Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2), 117 , 293-324 (1983). · Zbl 0531.57041 · doi:10.2307/2007079 |
[6] | Davis, M. W.: Nonpositive curvature and reflection groups. Handbook of Geometric Topology (eds. Daverman, R. J., and Sher, R. B.). North-Holland, Amsterdam, pp. 373-422 (2002). · Zbl 0998.57002 |
[7] | Dranishnikov, A. N.: On boundaries of hyperbolic Coxeter groups. Topology Appl., 110 (1), 29-38 (2001). · Zbl 0973.20030 · doi:10.1016/S0166-8641(99)00172-8 |
[8] | Ghys, E., and de la Harpe, P. (eds.): Sur les Groupes Hyperboliques d’après Mikhael Gromov. Progr. Math. vol. 83, Birkhäuser, Boston (1990). · Zbl 0731.20025 |
[9] | Hosaka, T.: Determination up to isomorphism of right-angled Coxeter systems. (2001). (Preprint). · Zbl 1054.20021 · doi:10.3792/pjaa.79.33 |
[10] | Humphreys, J. E.: Reflection groups and Coxeter groups. Cambridge Univ. Press, Cambridge-New York (1990). · Zbl 0725.20028 |
[11] | Moussong, G.: Hyperbolic Coxeter groups. Ph.D. Thesis, The Ohio State University (1988). |
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