scholarly journals Moufang affine buildings have Moufang spherical buildings at infinity

1997 ◽  
Vol 39 (3) ◽  
pp. 237-241 ◽  
Author(s):  
H. van Maldeghem ◽  
K. van Steen
Keyword(s):  
Spherical Building ◽  
Affine Buildings ◽  
Moufang Condition ◽  
Affine Building ◽  
Higher Rank

AbstractWe show in a direct and elementary way that the spherical building at infinity of every rank 3 affine building which satisfies Tits' Moufang condition, is itself a Moufang building. This result is also true for higher rank affine buildings by Tits' classification [4].

Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Simplicial Complex ◽  
Coxeter Groups ◽  
Fundamental Result ◽  
Spherical Building ◽  
Fundamental Properties ◽  
Moufang Condition ◽  
Affine Building ◽  
Tits Building ◽  
Basic Facts

This chapter assembles a few standard definitions, fixes some notation, and reviews a few of the results about buildings and Moufang polygons. It also summarizes the basic facts about Coxeter groups and buildings, including the fundamental properties of roots, residues, apartments, and projection maps. The chapter defines a Moufang building as spherical, thick, irreducible and of rank at least 2, and a Bruhat-Tits building as a thick irreducible affine building whose building at infinity is Moufang. Furthermore, it presents a fundamental result of Tits: that an irreducible thick spherical building of rank at least 3 satisfies the Moufang condition as do all the irreducible residues of rank at least 2 of such a building. Finally, it considers a simplicial complex, the dimension of which is its cardinality minus one.

scholarly journals A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

2019 ◽  
Vol 31 (5) ◽  
pp. 1317-1330
Author(s):  
Russell Ricks
Keyword(s):  
Symmetric Space ◽  
Group Action ◽  
Spherical Building ◽  
Rigidity Result ◽  
One Dimensional ◽  
Higher Rank ◽  
Geodesically Complete ◽  
Euclidean Building ◽  
Alternate Condition

AbstractWe prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

Galois Involutions

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Points ◽  
Polar Region ◽  
Arctic Region ◽  
The Arctic ◽  
Spherical Building ◽  
Linear Automorphism ◽  
Moufang Condition ◽  
Coxeter Diagram ◽  
Definition Of ◽  
The Arctic Region

This chapter focuses on the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions laid out in Hypothesis 30.1. It begins with the fhe definition of a spherical building satisfying the Moufang condition and a Galois involution of Δ‎, described as an automorphism of Δ‎ of order 2 that is strictly semi-linear. It can be recalled that Δ‎ can have a non-type-preserving semi-linear automorphism only if its Coxeter diagram is simply laced. The chapter assumes that the building Δ‎ being discussed is as in 30.1 and that τ‎ is a Galois involution of Δ‎. It also considers the notation stating that the polar region of a root α‎ of Δ‎ is the unique residue of Δ‎ containing the arctic region of α‎.

ENDOMORPHISMS OF DISCRETE GROUPS ACTING CHAMBER TRANSITIVELY ON AFFINE BUILDINGS

1993 ◽  
Vol 03 (03) ◽  
pp. 357-364 ◽  
Author(s):  
JOHN MEIER
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Discrete Groups ◽  
Affine Buildings ◽  
Locally Finite ◽  
Finite Affine ◽  
Affine Building

Let Γ be a group acting chamber transitively by type preserving automorphisms on a locally finite affine building of type Ã2. We show that Out(Γ) is finite and that Γ is Hopfian. We apply our results to affine Coxeter groups and a family of four groups discovered by J. Tits.

Pseudo-Split Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Spherical Building ◽  
Field Of Definition ◽  
Moufang Condition ◽  
Definition Of

This chapter introduces a class of Moufang spherical buildings known as pseudo-split buildings and considers the notion of the field of definition of a spherical building satisfying the Moufang condition. It begins with the notation: Let Δ‎ be an irreducible spherical building satisfying the Moufang condition, and let ℓ denote its rank (so ℓ is greater than or equal to 2 by definition). It then characterizes pseudo-split buildings as the spherical buildings which can be embedded in a split building of the same type. It also presents the proposition stating that every pseudo-split building is a subbuilding of a split building.

Moufang Structures

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Unipotent Group ◽  
Spherical Building ◽  
Unipotent Element ◽  
Moufang Condition ◽  
Descent Group

This chapter uses the notion of a Moufang structure to show that if Δ‎ is a spherical building satisfying the Moufang condition and Γ‎ is a descent group of Δ‎, then the fixed point building Δ‎Γ‎ also satisfies the Moufang condition. The discussion begins with the notation: Let (W, S) denote the type of Δ‎, let G = Aut(Δ‎) and let G° denote the group of type-preserving elements of G. The chapter then presents the conditions for an element g of G to be unipotent and for a subgroup U of G to be unipotent. It also describes a unipotent group U stabilizing a residue R and a unipotent element fixing two chambers x and y. Finally, it considers the set of extensions that forms a group acting faithfully on R.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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