Compactification of the Bruhat-Tits building of PGL by seminorms

Werner, Annette

doi:10.1007/s00209-004-0667-7

Cited by 17 publications

(56 citation statements)

References 9 publications

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“…We also plan to compare our approach to some concrete compactifications defined by A. Werner in the SL n case [Wer01], [Wer04]. In the present paper, the latter case is presented in the last section as an illustration of the general semisimple case.…”

Section: Introductionmentioning

confidence: 99%

Group-theoretic compactification of Bruhat–Tits buildings

Guivarc’h

Rémy

2006

Annales Scientifiques de l’École Normale Supérieure

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Let GF denote the rational points of a semisimple group G over a non-archimedean local field F , with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the BruhatTits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups, and the other is about subgroups with distal adjoint action.

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Section: Introductionmentioning

confidence: 99%

Group-theoretic compactification of Bruhat–Tits buildings

Guivarc’h

Rémy

2006

Annales Scientifiques de l’École Normale Supérieure

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“…This space was already considered in the case when k is local, in [Wer04]. We fix a finite dimensional vector space E over k. Given two norms n, n on E we denote…”

Section: The Space Of Norms and Seminormsmentioning

confidence: 99%

Almost algebraic actions of algebraic groups and applications to algebraic representations

Bader¹,

Duchesne²,

Lécureux³

2017

Groups Geom. Dyn.

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Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon [BF13].

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“…For the Bruhat-Tits building ∆ BT (G) of a reductive algebraic group over a local field, the corresponding Satake compactifications have been constructed in [151] [150] [171] [172] [173]. The paper [150] constructs the compactification by using Berkovich analytic geometry over complete non-Archimedean fields, and the paper [151] uses irreducible representations of the algebraic group and is more similar to the Satake compactifications of symmetric spaces.…”

Section: Proposition 625 (1) If ∆ Is a Euclidean Building Then Itsmentioning

confidence: 99%

“…The construction in [171] is also similar to the Satake compactifications of symmetric spaces. Compactifications of some special buildings were treated in [172] and [173].…”

Section: Proposition 625 (1) If ∆ Is a Euclidean Building Then Itsmentioning

confidence: 99%

Curve Complexes Versus Tits Buildings: Structures and Applications

2011

Buildings, Finite Geometries and Groups

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Abstract. Tits buildings ∆ Q (G) of linear algebraic groups G defined over the field of rational numbers Q have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of G(Q). Curve complexes C(Sg,n) of surfaces Sg,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the BruhatTits buildings of linear algebraic groups defined over local fields. In this chapter, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and at the same time suggest methods to solve them. Furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

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Compactification of the Bruhat-Tits building of PGL by seminorms

Cited by 17 publications

References 9 publications

Group-theoretic compactification of Bruhat–Tits buildings

Group-theoretic compactification of Bruhat–Tits buildings

Almost algebraic actions of algebraic groups and applications to algebraic representations

Curve Complexes Versus Tits Buildings: Structures and Applications

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