arXiv subjects

Koen Struyve

Publications and source records attributed to Koen Struyve.

23 records · Page 2Linked to original sources

arXiv2009

Two-dimensional affine R-buildings defined by generalized polygons with non-discrete valuation

In this paper we complete the proof of the "equivalence" of non-discrete R-buildings of types A~_2 and C~_2, with, respectively, projective planes and generalized quadrangles with non-discrete valuation, begun in previous paper of the authors. We also complete the proof of the "equivalence" of an affine building of rank 3 with a generalized polygon with discrete valuation (by proving this for generalized hexagons). We also complement a result of the second author by proving uniqueness up to scalar multiples of the weight sequences of polygons with non-discrete valuation. As an application, we produce some new explicitly defined non-discrete R-buildings, in particular a class of type A~_2 with arbitrary residues.

math.MG↗

arXiv2009

(Non-)completeness of R-buildings and fixed point theorems

We prove two generalizations of results proved by Bruhat and Tits involving metrical completeness and R-buildings. Firstly, we give a generalization of the Bruhat-Tits fixed point theorem also valid for non-complete R-buildings, but with the added condition that the group is finitely generated. Secondly, we generalize a criterion which reduces the problem of completeness to the wall trees of the R-building.

math.MG↗

arXiv2009

Moufang quadrangles of mixed type

In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual net associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in an already known result by the second author.

math.CO↗

arXiv2009

Ree geometries

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines.

math.GR↗

arXiv2009

Moufang sets related to polarities in exceptional Moufang quadrangles of type F_4

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F_4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type F_4 stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang n-gons.

math.GR↗