Abstract
We prove an affine analog of Scharlau’s reduction theorem for spherical buildings. To be a bit more precise let X be a euclidean building with spherical building \(\partial X\) at infinity. Then there exists a euclidean building \(\bar{X}\) such that X splits as a product of \(\bar{X}\) with some euclidean k-space such that \(\partial \bar{X}\) is the thick reduction of \(\partial X\) in the sense of Scharlau. In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.
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References
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)
Bennett, C.D., Schwer, P.: On axiomatic definitions of non-discrete affine buildings. (Appendix by Koen Struyve). Adv. Geom. 14(3), 381–412 (2014)
Bennett, C.D.: Affine \(\Lambda \)-buildings. Dissertation, Chicago Illinois, pp. 1–106 (1990)
Brown, K.S.: Buildings. Springer, New York (1989)
Caprace, P.-E.: The thick frame of a weak twin building. Adv. Geom. 5(1), 119–136 (2005)
Charney, R., Lytchak, A.: Metric characterizations of spherical and euclidean buildings. Geom. Topol. 5(2), 521–550 (2001)
Hitzelberger, P.: Generalized affine buildings. Doctoral Thesis, Münster, Germany, pp. 1–99. arXiv:0902.1107 (2009)
Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Publ. Math. IHES 86, 115–197 (1997)
Kramer, L.; Notes on completely reducible subcomplexes of spherical buildings. arXiv:1010.0083 (2010)
Parreau, A.: Immeubles affines: construction par les normes. In: Crystallographic Groups and Their Generalizations: Workshop, Katholieke Universiteit Leuven Campus Kortrijk, Belgium, May 26–28, 1999, vol. 262, p. 263. American Mathematical Soc., Providence (2000)
Rees, S.: Weak buildings of spherical type. Geom. Dedicata 27(1), 15–47 (1988)
Scharlau, R.: A structure theorem for weak buildings of spherical type. Geom. Dedicata 24(1), 77–84 (1987)
Tits, J.: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol. 386. Springer, Berlin (1974)
Tits, J.: Endliche Spiegelungsgruppen, die als Weylgruppen auftreten. Invent. Math. 43(3), 283–295 (1977)
Tits, J.: Immeubles de Type Affine. In: Buildings and the Geometry of Diagrams, Lect. 3rd 1984 Sess. C.I.M.E., Como/Italy 1984, Lect. Notes Math. Vol. 1181. Springer, Berlin (1986)
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We would like to thank the anonymous referee for many helpful comments and suggestions. The first author would like to thank Misha Kapovich for helpful discussions.
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Schwer, P., Weniger, D. A structure theorem for euclidean buildings. J. Geom. 110, 18 (2019). https://doi.org/10.1007/s00022-019-0473-3
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DOI: https://doi.org/10.1007/s00022-019-0473-3