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Right-angled mock reflection and mock Artin groups


Author: Richard Scott
Journal: Trans. Amer. Math. Soc. 360 (2008), 4189-4210
MSC (2000): Primary 20Fxx
DOI: https://doi.org/10.1090/S0002-9947-08-04452-8
Published electronically: March 12, 2008
MathSciNet review: 2395169
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Abstract: We define a right-angled mock reflection group to be a group $ G$ acting combinatorially on a CAT(0) cubical complex such that the action is simply-transitive on the vertex set and all edge-stabilizers are $ \mathbb{Z}_2$. We give a combinatorial characterization of these groups in terms of graphs with local involutions. Any such graph $ \Gamma$ not only determines a mock reflection group, but it also determines a right-angled mock Artin group. Both classes of groups generalize the corresponding classes of right-angled Coxeter and Artin groups. We conclude by showing that the standard construction of a finite $ K(\pi,1)$ space for right-angled Artin groups generalizes to these mock Artin groups.


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  • 1. M. Bridson and A. Haefliger. Metric Spaces of Non-positive Curvature. Springer-Verlag, Berlin, Heidelberg, and New York, 1999. MR 1744486 (2000k:53038)
  • 2. R. Charney and M. Davis. Finite $ K(\pi,1)$'s for Artin Groups. Prospects in topology, Ann. of Math. Stud., 138, Princeton Univ. Press (1995), 110-124. MR 1368655 (97a:57001)
  • 3. M. Davis. Nonpositive curvature and reflection groups. Handbook of Geometric Topology, eds. R. Daverman and R. Sher, North-Holland, Amsterdam (2002), 373-422. MR 1886674 (2002m:53061)
  • 4. M. Davis, T. Januszkiewicz, and R. Scott. Nonpositive curvature of blow-ups. Sel. math., New ser. 4 (1998), 491-547. MR 1668119 (2001f:53078)
  • 5. M. Davis, T. Januszkiewicz, and R. Scott. Fundamental groups of blow-ups. Adv. Math. 177 (2003), 115-179. MR 1985196 (2004c:57030)
  • 6. P. Etingof, A. Henriques, J. Kamnitzer, and E. Rains. The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. math.AT/0507514.
  • 7. M. Kapranov. The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equations. J. Pure and Applied Algebra 85 (1993), 119-142. MR 1207505 (94b:52017)
  • 8. M. Kapranov. Chow quotients of Grassmannians. I. Adv. in Sov. Math. 16 (1993), 29-110. MR 1237834 (95g:14053)
  • 9. R. Lyndon and P. Schupp. Combinatorial Group Theory. Springer-Verlag, Berlin, Heidelberg, and New York, 1977. MR 0577064 (58:28182)
  • 10. A. Haefliger. Complexes of groups and orbihedra. Group Theory from a Geometrical Viewpoint (A. Ghys, A. Haefliger and A. Verjovsky, eds.), World Scientific Publishing Co., River Edge, NJ, 1991. MR 1170362 (93a:20001)
  • 11. M. Sageev. Ends of group pairs and non-positively curved cube complexes. Proc. London Math. Soc. 71 (1995), no. 3, 585-617. MR 1347406 (97a:20062)

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Additional Information

Richard Scott
Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053
Email: rscott@math.scu.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04452-8
Received by editor(s): June 26, 2006
Published electronically: March 12, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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