On topological twin buildings and topological split Kac-Moody groups

Hartnick, Tobias; Köhl, Ralf; Mars, Andreas

doi:10.48550/arxiv.1201.3562

Cited by 1 publication

(15 citation statements)

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“…They will be proved among other results given in Sections 3.2 and 3.3 to which we refer for the details. As already indicated above, these results provide a link to previous contributions of Kramer [Kra02] and Hartnick-Köhl-Mars [HKM13] on topological twin buildings. Especially in the latter reference the authors discuss several axioms that one could require for a topological twin building.…”

Section: The Cone Topology On a Masuresupporting

confidence: 86%

“…In Section 3 we define and study the cone topology on the twin building at infinity of a masure. We prove Theorem 1.1 and several other results which are motivated by the axioms for topological twin buildings discussed in [HKM13].…”

Section: Organization Of the Papermentioning

confidence: 71%

“…In that paper, topological twin buildings are constructed by means of the Kac-Peterson topology of a Kac-Moody group. It turns out that masures provide examples of weak topological twin buildings and this construction appears to be much more natural and direct than the one given in Hartnick-Köhl-Mars [HKM13].…”

Section: Introductionmentioning

confidence: 89%

“…For this reason we provide several basic results regarding the cone topology which show that it has several remarkable properties. As a consequence of our investigation, it turns out that the twin building at infinity of a masure endowed with the cone topology is a "weak topological twin building" in the sense of Hartnick-Köhl-Mars [HKM13]. In that paper, topological twin buildings are constructed by means of the Kac-Peterson topology of a Kac-Moody group.…”

Section: Introductionmentioning

confidence: 89%

“…We write δ and δ * its distance and codistance. Recall the following piece of notation, classical or issued from Hartnick-Köhl-Mars [HKM13].…”

Section: Twin Buildingsmentioning

confidence: 99%

See 4 more Smart Citations

The cone topology on masures

Ciobotaru

Mühlherr

Rousseau

2020

Advances in Geometry

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Masures are generalizations of Bruhat-Tits buildings and the main examples are associated with almost split Kac-Moody groups G over non-Archimedean local fields. In this case, G acts strongly transitively on its corresponding masure ∆ as well as on the building at infinity of ∆, which is the twin building associated with G. The aim of this article is twofold: firstly, to introduce and study the cone topology on the twin building at infinity of a masure. It turns out that this topology has various favorable properties that are required in the literature as axioms for a topological twin building. Secondly, by making use of the cone topology, we study strongly transitive actions of a group G on a masure ∆. Under some hypotheses, with respect to the masure and the group action of G, we prove that G acts strongly transitively on ∆ if and only if it acts strongly transitively on the twin building at infinity ∂∆. Along the way a criterion for strong transitivity is given and the existence and good dynamical properties of strongly regular hyperbolic automorphisms of the masure are proven. * corina.ciobotaru@unifr.ch, Partially supported by Swiss National Science Foundation Grant 153599 † Bernhard.M.Muehlherr@math.uni-giessen.de ‡ Guy.Rousseau@univ-lorraine.fr § auguste.hebert@univ-st-etienne.fr, Partially supported by ANR grant ANR-15-CE40-0012 J∈Σ Res J (X ± ). Take (res J (d)) J∈Σ for someConsider r d such that r ∈ (x, r d ). We can suppose that because d ∈ U x,r,c . LetLet A ′ be the apartment of ∆ containing c − and d at infinity. Actually, by Lemma 3.8, the closed sectors Q x,c − and Q x,d are opposite and in A ′ .Recall that for a spherical face φ at infinity of ∆ and a point z ∈ ∆ there exists a uniquely defined sector face Q z,φ with base point z and ideal face φ. This might not be the case when J ⊂ S is not spherical.For J ∈ Σ, let φ J (resp., φ ′ J ) be the ideal face of type J of d (resp., d ′ or d J )., for any Euclidean metric on A ′ , the apartment A ′ contains any given bounded subset of QBy the hypothesis on Σ, the union J∈Σ Q x,φ J ⊂ Q x,d is in no wall; so the convex hull ofis big in Q x,d . By the axiom (MAO), this implies that Q x,d ∩ Q x,d ′ is big in Q x,d . So d ′ ∈ U x,r,c , for r d far enough.

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Section: The Cone Topology On a Masuresupporting

confidence: 86%

Section: Organization Of the Papermentioning

confidence: 71%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 89%

“…We write δ and δ * its distance and codistance. Recall the following piece of notation, classical or issued from Hartnick-Köhl-Mars [HKM13].…”

Section: Twin Buildingsmentioning

confidence: 99%