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Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings

Koen Struyve (UGent) and Hendrik Van Maldeghem (UGent)
(2010) ADVANCES IN GEOMETRY. 10(3). p.465-476
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Abstract
In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type (G) over tilde (2), and it reproves the results obtained mainly by the second author for types (A) over tilde (2) and (C) over tilde (2). The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building).
Keywords
TRIANGLE BUILDINGS, QUADRANGLES, valuations, generalized polygons, Affine buildings, systems of apartments

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Citation

Please use this url to cite or link to this publication:

Chicago
Struyve, Koen, and Hendrik Van Maldeghem. 2010. “Generalized Polygons with Non-discrete Valuation Defined by Two-dimensional Affine R-buildings.” Advances in Geometry 10 (3): 465–476.
APA
Struyve, Koen, & Van Maldeghem, H. (2010). Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings. ADVANCES IN GEOMETRY, 10(3), 465–476.
Vancouver
1.
Struyve K, Van Maldeghem H. Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings. ADVANCES IN GEOMETRY. 2010;10(3):465–76.
MLA
Struyve, Koen, and Hendrik Van Maldeghem. “Generalized Polygons with Non-discrete Valuation Defined by Two-dimensional Affine R-buildings.” ADVANCES IN GEOMETRY 10.3 (2010): 465–476. Print.
@article{1027951,
  abstract     = {In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type (G) over tilde (2), and it reproves the results obtained mainly by the second author for types (A) over tilde (2) and (C) over tilde (2). The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building).},
  author       = {Struyve, Koen and Van Maldeghem, Hendrik},
  issn         = {1615-715X},
  journal      = {ADVANCES IN GEOMETRY},
  keyword      = {TRIANGLE BUILDINGS,QUADRANGLES,valuations,generalized polygons,Affine buildings,systems of apartments},
  language     = {eng},
  number       = {3},
  pages        = {465--476},
  title        = {Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings},
  url          = {http://dx.doi.org/10.1515/ADVGEOM.2010.018},
  volume       = {10},
  year         = {2010},
}

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