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Register a new userNotes on completely reducible subcomplexes of spherical buildings
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Authors
Linus Kramer
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A completely reducible subcomplex of a spherical building is a spherical building.
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In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise $ell^infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural piecewise $ell^infty$ metric which is coarsely Helly. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. The only exception is the special linear group: if $n geq 3$ and $mathbb{K}$ is a local field, we show that $operatorname{SL}(n,mathbb{K})$ does not act properly and coboundedly on an injective metric space.
We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.
This is a survey on nondiscrete euclidean buildings, with a focus on metric properties of these spaces.
We construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to Bruhat-Tits buildings, for which we also provide a Tannakian description.
Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.
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