(Non)-completeness of $ℝ$-buildings and fixed point theorems

Struyve, Koen

doi:10.4171/ggd/121

Cited by 11 publications

(8 citation statements)

References 13 publications

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“…Struyve recently proved the following generalization of the Bruhat-Tits Fixed Point Theorem. If a finitely generated group acts acts isometrically and with bounded orbits on a euclidean building, then it has a fixed point [32]. Moreover, he showed that the main rigidity results in [22] also hold if the completeness assumptions on the euclidean buildings are dropped (unpublished).…”

Section: Lemmamentioning

confidence: 99%

Metric Properties of Euclidean Buildings

Krämer

2011

Global Differential Geometry

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This is a survey on nondiscrete euclidean buildings, with a focus on metric properties of these spaces.Euclidean buildings are higher dimensional generalizations of trees. Indeed, the euclidean product X of two (leafless) metric trees T 1 , T 2 is already a good "toy example" of a 2-dimensional euclidean building. The space X contains lots of copies of the euclidean plane E 2 and has at the same time a complicated local branching.Euclidean building were invented by Jacques Tits in the seventies. Similarly as in the case of spherical buildings, their definition was motivated by group theoretic questions. While spherical buildings are by now a standard tool in the structure theory of reductive algebraic groups over arbitrary fields, euclidean buildings are important for the advanced structure theory of reductive groups over fields with valuations. In particular, they are very much linked to number theory and arithmetic geometry.In the last 25 years, however, euclidean buildings have also become important in geometry. This is due to the fact that euclidean buildings are spaces of nonpositive curvature. But more is true. Together with the Riemannian symmetric spaces of nonpositive curvature, euclidean buildings could be called the islands of high symmetry in the world of CAT(0) spaces. This claim will be made more precise below. Almost inevitably, questions about symmetry, rigidity, or higher rank for CAT(0) spaces lead to these geometries.I thank Alexander Lytchak, Koen Struyve and Richard Weiss for helpful comments.

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Section: Lemmamentioning

confidence: 99%

Metric Properties of Euclidean Buildings

Krämer

2011

Global Differential Geometry

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“…A special case of the Center Theorem has been applied in geometric invariant theory; see [11], [14, p.64] and [18]. Another, more recent, application of the Center Theorem has been obtained by K. Struyve who used it in [20] to obtain a fixed point theorem for finitely generated bounded groups of isometries acting on affine R-buildings. This fixed point theorem validiated many results in the thesis of G. Rousseau [17], an important ANNALES DE L'INSTITUT FOURIER contribution to Bruhat-Tits theory.…”

Section: Introductionmentioning

confidence: 99%

Receding polar regions of a spherical building and the center conjecture

Mühlherr¹,

Weiss²

2013

Ann. inst. Fourier

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“…Proof. By [26,Lemma 4.4] (which uses results from [9]) the R-building X can be isometrically embedded in a metrically complete R-building X ′ of the same type, such that the G-action on X extends to X ′ . Suppose G fixes a point of X ′ .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%