Λ-buildings and base change functors

Schwer, Petra; Struyve, Koen

doi:10.1007/s10711-011-9611-2

Cited by 7 publications

(5 citation statements)

References 13 publications

(21 reference statements)

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“…Let Λ, Λ ′ be totally ordered abelian groups and e : Λ → Λ ′ be a morphism of ordered groups. Then Schwer and Struyve construct a functor from the category of Λ-buildings to the category of Λ ′ -buildings, compatible with e (see [SS12]). Using this and using ultraproducts, they construct nontrivial examples of Λ-buildings, for Λ R. They in particular construct ultracones and asymptotic cones of buildings (see [SS12, Section 6]).…”

Section: Bennett's λ-Buildingsmentioning

confidence: 99%

$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

Hébert,

Izquierdo,

Loisel

2020

Preprint

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Let G be a quasi-split reductive group and K be a Henselian field equipped with a valuation ω : K × → Λ, where Λ is a totally ordered abelian group. In 1972, Bruhat and Tits constructed a building on which the group G(K) acts provided that Λ is a subgroup of R. In this paper, we deal with the general case where there are no assumptions on Λ and we construct a set on which G(K) acts. We then prove that it is a Λ-building, in the sense of Bennett.

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Section: Bennett's λ-Buildingsmentioning

confidence: 99%

$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

Hébert,

Izquierdo,

Loisel

2020

Preprint

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“…The coordinate changes are now described by the nonstandard affine Weyl group W ⋉ * R n . One can show that X D is a generalized affine building in the sense of Bennett [3] [20] [43]. We call X D a nonstandard Euclidean building, with nonstandard charts, nonstandard apartments, nonstandard Weyl simplices, and so on.…”

Section: Ultraproducts Of Euclidean Buildingsmentioning

confidence: 99%

Coarse equivalences of Euclidean buildings

Krämer

Weiss

2014

Advances in Mathematics

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We prove the following rigidity results. Coarse equivalences between metrically complete Euclidean buildings preserve spherical buildings at infinity. If all irreducible factors have dimension at least two, then coarsely equivalent Euclidean buildings are isometric (up to scaling factors); if in addition none of the irreducible factors is a Euclidean cone, then the isometry is unique and has finite distance from the coarse equivalence.The appendix shows how these results can be extended to non-complete Euclidean buildings.We prove coarse (i.e. quasi-isometric) rigidity results for leafless trees (simplicial trees and R-trees with extensible geodesics) and, more generally, for discrete and nondiscrete Euclidean buildings. For trees, a key ingredient is a certain equivariance condition. Our main results are as follows.Theorem I Let G be a group acting isometrically on two metrically complete leafless trees T 1 , T 2 . Assume that there is a coarse equivalence f : T 1 ✲ T 2 , that T 1 has at least 3 ends and that the induced map ∂f : ∂T 1 ✲ ∂T 2 between the ends of the trees is G-equivariant. If the G-action on ∂T 1 is 2-transitive, then (after rescaling the metric on T 2 ) there is a G-equivariant isometryf : T 1 ✲ T 2 with ∂f = ∂f . If T 1 has at least two branch points, thenf is unique and has finite distance from f .The precise result is 2.18 below, where we also consider products of trees and Euclidean spaces. Without the equivariance condition, quasi-isometric rigidity fails. The letters X and H written infinitely large are, for example, trees which are coarsely equivalent without being isometric. Note, too, that any two simplicial trees of finite constant valence are coarsely equivalent without being necessarily isometric [34].Our next results are concerned with (possibly nondiscrete) Euclidean buildings.Theorem II Let X 1 and X 2 be Euclidean buildings whose spherical buildings at infinity ∂ cpl X 1 and ∂ cpl X 2 are thick. Suppose that f : X 1 × R m 1 ✲ X 2 × R m 2 is a coarse equivalence. Then m 1 = m 2 and there is a combinatorial isomorphism f * : ∂ cpl X 1 ✲ ∂ cpl X 2 between the spherical buildings at infinity which is characterized by the fact that for an affine apartment A ⊆ X 1 the f -image of A × R m 1 has finite Hausdorff distance from f * (A) × R m 2 . This is 5.16 below. We remark that the boundary map f * is constructed in a combinatorial way from f . In general, a coarse equivalence between CAT(0)-spaces will not induce a map between the respective Tits boundaries.Theorem III Let f : X 1 × R m ✲ X 2 × R m be as in Theorem II and assume in addition that X 1 has no tree factors and that X 1 and X 2 are metrically complete. Then there is (possibly after rescaling * Kramer and Weiss were supported by the SFB 878 and DFG Project KR 1668/7. 1 the metrics on the de Rham factors of X 2 ) an isometryf :If none of the de Rham factors of X 1 is a Euclidean cone over its boundary, thenf is unique and d(f 1 (x × y),f (x)) is bounded as a function of x ∈ X 1 .For a more general statement see 5.21 b...

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“…The only conditions one still has to verify are part of Condition (A2) and Condition (A3). The strategy we apply here is the same as used by Petra N. Schwer and the second author in [22]. This section only uses Condition (A5) and no hypothesis on the minimal angle.…”

Section: Conditions (A2) and (A3)mentioning

confidence: 99%

Descent of affine buildings - I. Large minimal angles

Mühlherr¹,

Struyve

Maldeghem

2014

Trans. Amer. Math. Soc.

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In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits.This first part lays the foundations for our approach and deals with the 'large minimal angle' case.

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Λ-buildings and base change functors

Cited by 7 publications

References 13 publications

$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

Coarse equivalences of Euclidean buildings

Descent of affine buildings - I. Large minimal angles

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