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Twin trees and $\lambda _{\Lambda }$-gons

Author: Curtis D. Bennett
Journal: Trans. Amer. Math. Soc. 349 (1997), 2069-2084
MSC (1991): Primary 51E12, 20E99
MathSciNet review: 1370635
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Abstract: We define a natural generalization of generalized $n$-gons to the case of $\Lambda $-graphs (where $\Lambda $ is a totally ordered abelian group and $0<\lambda \in \Lambda$). We term these objects $\lambda _{\Lambda }$-gons. We then show that twin trees as defined by Ronan and Tits can be viewed as $(1,0)_{\Lambda }$-gons, where $\Lambda = Z \times Z$ is ordered lexicographically. This allows us to then generalize twin trees to the case of $\Lambda $-trees. Finally, we give a free construction of $\lambda _{\Lambda }$-gons in the cases where $\Lambda $ is discrete and has a subgroup of index $2$ that does not contain the minimal element of $\Lambda $.

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  • [AB] R. Alperin and H. Bass, Length Functions of Group Actions on $\Lambda $-trees, Proceedings of the 1984 Alta Conference on Combinatorial Group Theory and Topology, Annals of Mathematical Studies, vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265-378. MR 89c:20057
  • [B] S.A. Basarab, On a Problem Raised by Alperin and Bass, in Arboreal Group Theory, (ed. R. Alperin), Mathematical Sciences Research Institute Publications 19, Springer-Verlag, New York, 1991, pp. 35-68. MR 92b:57001
  • [Be] C. Bennett, Generalized Spherical Buildings, preprint.
  • [K] W. Kantor, Generalized Polygons, SCABs and GABs, in Buildings and the Geometry of Diagrams (ed. L.A. Rosati), Lecture Notes in Mathematics 1181, Springer-Verlag, Berlin, 1986, pp. 79-158. MR 87k:51014
  • [MS] J. Morgan and P. Shalen, Valuations, Trees, and Degenerations of Hyperbolic Structures, I, Annals of Mathematics 120 (1984), 401-476. MR 86f:57011
  • [R] M. Ronan, Lectures on Buildins, Academic Press, San Diego (1989). MR 80j:20001
  • [RT] M. Ronan and J. Tits, Twin Trees, I, Inventiones Mathematicae 116 (1994), 463-479. MR 94k:20058.
  • [S] J-P. Serre, Trees, Springer-Verlag, Berlin, 1980. MR 82c:20083
  • [T1] J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, 386, Springer-Verlag, Berlin (1974). MR 57:9866
  • [T2] J. Tits, Course Notes on Twin Buildings Annuaire du Collège de France, 89e année, 1988-1989, 81-95.
  • [T3] J. Tits, Course Notes on Twin Buildings Annuaire du Collège de France, 90e année, 1989-1990, 87-103.
  • [T4] J. Tits, Twin Buildings and Groups of Kac-Moody Type, in Groups, Combinatorics, and Geometry, Durham 1990 (ed. M. Liebeck and J. Saxl), London Mathematical Society Lecture Notes 165, Cambridge University Press, 1992, pp. 249-286. MR 94d:20030

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Additional Information

Curtis D. Bennett
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403

Keywords: $\Lambda$-trees, twin trees, generalized $n$-gons
Received by editor(s): April 24, 1994
Received by editor(s) in revised form: January 4, 1996
Additional Notes: The author gratefully acknowledges the support of an NSF postdoctoral fellowship.
Article copyright: © Copyright 1997 American Mathematical Society

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