The cone topology on masures

Abstract

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.

Macdonald's formula for Kac–Moody groups over local fields

“…First in the case n = 0. (11) holds true in this case! Second, the case n > 0 and s 0 = s 1 reads as follows, taking into account that…”

“…(4) We assume that there is a group acting strongly transitively on (by automorphisms), that is, all isomorphisms involved in (1a) or (1c) above are induced by elements of , cf . [; , 4.10]. The main example is when is an almost split Kac–Moody group over a non‐Archimedean local field and is its masure, see 1.12.…”

“…all isomorphisms involved in 1.a or 1.c above are induced by elements of G, cf. [Ro17, 4.10] and [CiMR17]. We choose in I a fundamental apartment which we identify with A.…”

Martin compactifications of affine buildings

“…this is part of the content of [18,Theorem 2.8] in which more precise statements can be found. Remark 1.…”

“…Let S be a sector one of whose sector faces represents ∞ : this means that ∞ is a facet of ∞ S, or also that the parallelism class of S, denoted by [S] / / , belongs to ( ∞ ). The sectors in X ( ∞ ) are the images under the above map ∞ of such sectors [18,Section 2.6]. Let S and S ′ be two such sectors.…”