Algebraic groups with few subgroups

Garibaldi, Skip; Gille, Philippe

doi:10.1112/jlms/jdp030

Cited by 20 publications

(11 citation statements)

References 40 publications

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“…Remark. In case char k = 2, one can use the Rost invariant to define a class r(G) ∈ H 3 (k, Z/2Z) depending only on G, see [GaGi,§7]. If L/k is an extension such that X 2 (L) is nonempty, then certainly r(G) is killed by L, hence [L : k]r(G) = 0.…”

Section: Applications To the Rost Invariantmentioning

confidence: 99%

Shells of twisted flag varieties and the Rost invariant

Garibaldi¹,

Petrov

Semenov

2016

Duke Math. J.

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Section: Applications To the Rost Invariantmentioning

confidence: 99%

Shells of twisted flag varieties and the Rost invariant

Garibaldi¹,

Petrov

Semenov

2016

Duke Math. J.

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“…In particular the hypotheses are satisfied for Q-structures on G = SL n (R) for n prime ( [GG,Prop. 4.1]).…”

Section: Construction Of a Dense Forestmentioning

confidence: 99%

Dense forests and Danzer sets

Solomon¹,

Weiss²

2016

Ann. Sci. École Norm. Sup.

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Abstract. A set Y ⊆ R d that intersects every convex set of volume 1 is called a Danzer set. It is not known whether there are Danzer sets in R d with growth rate O(T d ). We prove that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets. For cut and project sets our proof relies on the dynamics of homogeneous flows. We consider a weakening of the Danzer problem, the existence of a uniformly discrete dense forests, and we use homogeneous dynamics (in particular Ratner's theorems on unipotent flows) to construct such sets. We also prove an equivalence between the above problem and a well-known combinatorial problem, and deduce the existence of Danzer sets with growth rate O(T d log T ), improving the previous bound of O(T d log d−1 T ).

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“…Under the additional hypothesis that

m

is not divisible by

char k

, this result was proved in [, Proposition 7.2] using the elementary theory of cohomological invariants from . We give a proof valid for all characteristics that relies on the (deeper) theory of invariants of degree 3 of semisimple groups developed in .…”

Section: Generalitiesmentioning

confidence: 87%

Tits p‐indexes of semisimple algebraic groups

Clercq

Garibaldi

2017

J. London Math. Soc.

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Abstract. The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria of motivic equivalence for semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants and Chow motives of the twisted flag varieties for those groups.The Tits index (sometimes called Satake diagram) of a semisimple linear algebraic group G over a field k includes as special cases the classical notions of Schur index of a central simple associative algebra and the Witt index of a quadratic form. It is a fundamental invariant of semisimple algebraic groups. However, for the purpose of stating and proving theorems about Chow motives with F p coefficients, one should consider not the Tits index of G, but rather the (Tits) p-index, meaning the Tits index of G L where L is an algebraic extension of k of degree not divisible by p, yet all the finite algebraic extensions of L have degree a power of p. Such an L is called a p-special closure of k in [8, §101.B] and all such fields are isomorphic as k-algebras, so the notion of Tits p-index over k is well defined.Let G be a semisimple algebraic group over k. As shown in [7], the Tits p-indexes of G on all fields extensions of k -the higher Tits p-indexes of G -determine the motivic equivalence class of G modulo p. The aim of this article is to determine the values of the Tits p-indexes of the absolutely simple algebraic groups, using as a starting point the known list of possible Tits indexes as in [45], [43], or [36]. Along the way, we give in some cases criteria for motivic equivalence for semisimple groups in terms of their algebraic and cohomological invariants.Date: 10 novembre 2015.

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Algebraic groups with few subgroups

Cited by 20 publications

References 40 publications

Shells of twisted flag varieties and the Rost invariant

Shells of twisted flag varieties and the Rost invariant

Dense forests and Danzer sets

Tits p‐indexes of semisimple algebraic groups

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