Strong Transitivity, the Moufang Condition and the Howe–Moore Property

Corina-Gabriela Ciobotaru

doi:10.1007/s00031-022-09766-0

Strong Transitivity, the Moufang Condition and the Howe–Moore Property

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building Δ of dimension ≥ 2 that acts strongly transitively on Δ is Moufang. If moreover Δ is irreducible and H is topologically simple, we show that H is the subgroup G(k) ⁺ of the k-rational points G(k) of the isotropic simple algebraic group G over a non-Archimedean local field k associated with Δ. Secondly, we generalise the proof given in Burger and Mozes (Inst. Hautes Études Sci. Publ. Math., 92, 151–194 (2001) 2000) for the case of bi-regular trees to any locally finite thick affine building Δ, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(Δ) has the Howe–Moore property. This proof is different than the strategy used so far in the literature and does not rely on the polar decomposition KA ⁺K, where K is a maximal compact subgroup, and the important fact that A ⁺ is an abelian maximal sub-semi-group.

Originalsprog	Engelsk
Tidsskrift	Transformation Groups
Vol/bind	30
Nummer	1
Sider (fra-til)	165-185
Antal sider	21
ISSN	1083-4362
DOI	https://doi.org/10.1007/s00031-022-09766-0
Status	Udgivet - mar. 2025

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abstract = "Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building Δ of dimension ≥ 2 that acts strongly transitively on Δ is Moufang. If moreover Δ is irreducible and H is topologically simple, we show that H is the subgroup G(k) + of the k-rational points G(k) of the isotropic simple algebraic group G over a non-Archimedean local field k associated with Δ. Secondly, we generalise the proof given in Burger and Mozes (Inst. Hautes {\'E}tudes Sci. Publ. Math., 92, 151–194 (2001) 2000) for the case of bi-regular trees to any locally finite thick affine building Δ, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(Δ) has the Howe–Moore property. This proof is different than the strategy used so far in the literature and does not rely on the polar decomposition KA +K, where K is a maximal compact subgroup, and the important fact that A + is an abelian maximal sub-semi-group.",

author = "Corina-Gabriela Ciobotaru",

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T1 - Strong Transitivity, the Moufang Condition and the Howe–Moore Property

AU - Ciobotaru, Corina-Gabriela

PY - 2025/3

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N2 - Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building Δ of dimension ≥ 2 that acts strongly transitively on Δ is Moufang. If moreover Δ is irreducible and H is topologically simple, we show that H is the subgroup G(k) + of the k-rational points G(k) of the isotropic simple algebraic group G over a non-Archimedean local field k associated with Δ. Secondly, we generalise the proof given in Burger and Mozes (Inst. Hautes Études Sci. Publ. Math., 92, 151–194 (2001) 2000) for the case of bi-regular trees to any locally finite thick affine building Δ, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(Δ) has the Howe–Moore property. This proof is different than the strategy used so far in the literature and does not rely on the polar decomposition KA +K, where K is a maximal compact subgroup, and the important fact that A + is an abelian maximal sub-semi-group.

AB - Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building Δ of dimension ≥ 2 that acts strongly transitively on Δ is Moufang. If moreover Δ is irreducible and H is topologically simple, we show that H is the subgroup G(k) + of the k-rational points G(k) of the isotropic simple algebraic group G over a non-Archimedean local field k associated with Δ. Secondly, we generalise the proof given in Burger and Mozes (Inst. Hautes Études Sci. Publ. Math., 92, 151–194 (2001) 2000) for the case of bi-regular trees to any locally finite thick affine building Δ, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(Δ) has the Howe–Moore property. This proof is different than the strategy used so far in the literature and does not rely on the polar decomposition KA +K, where K is a maximal compact subgroup, and the important fact that A + is an abelian maximal sub-semi-group.

U2 - 10.1007/s00031-022-09766-0

DO - 10.1007/s00031-022-09766-0

M3 - Journal article

SN - 1083-4362

VL - 30

SP - 165

EP - 185

JO - Transformation Groups

JF - Transformation Groups

IS - 1

ER -

Strong Transitivity, the Moufang Condition and the Howe–Moore Property

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