Automorphisms of<i>p</i>–compact groups and their root data

Andersen, Kasper K. S.; Grodal, Jesper

doi:10.2140/gt.2008.12.1427

Cited by 3 publications

(20 citation statements)

References 31 publications

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“…11.3] is an elementary abelian 2-subgroup of W . Now, recall Tits' model for N G (T ) from [Tit66], elaborated and extended to 2-compact groups in [DW05] and [AG08]: N G (T ) can be constructed from the root datum by first constructing the reflection extension 1 → Z[Σ] → ρ(W ) → W → 1, where Σ is the set of reflections in W , and then constructing N G (T ) as a push-forward along a W -map f : Z[Σ] → T sending each reflection σ to a certain element of order two h σ of T ; see [AG08, § §2-3]. Let ρ 0 denote the preimage of W 0 in ρ(W ), and consider the subgroup A of N G (T ) generated by 2 T and the image of ρ 0 under ρ(W ) → N G (T ), the map to the push-forward.…”

Section: On the Existence Of A Fundamental Class: Proof Of Theorem Dmentioning

confidence: 99%

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String topology of finite groups of Lie type

Grodal,

Lahtinen

2020

Preprint

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We show that the mod cohomology of any finite group of Lie type in characteristic p = admits the structure of a module over the mod cohomology of the free loop space of the classifying space BG of the corresponding compact Lie group G, via ring and module structures constructed from string topology, à la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is nontrivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration.We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over Fq, with q congruent to 1 mod . We also show how to deal with twistings and get rid of the congruence condition by replacing BG by a certain -compact fixed point group depending on the order of q mod , without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings. Contents 1. Introduction 1 2. Construction of the products 7 3. Pairings on Serre spectral sequences 28 4. Proofs of Theorems A-C 32 5. On the existence of a fundamental class: Proof of Theorem D 35 6. On the existence of a fundamental class for general σ: Proof of Theorem E 40 Appendix A. Parametrized homotopy theory 45 Appendix B. Dualizability in a symmetric monoidal category 49 Appendix C. Z -root data and untwisting of finite groups of Lie type 50 References 56

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Section: On the Existence Of A Fundamental Class: Proof Of Theorem Dmentioning

confidence: 99%

“…B and C], which explain exactly how ψ q acts on N G (T ), namely as a quotient of a map which multiplies by q on T and is the identity on ρ(W ) (see Step 2 of the proof of Thm. B in [AG08] for the definition of the homomorphism s : Out(D G ) → Out(N G (T ))).…”

Section: On the Existence Of A Fundamental Class: Proof Of Theorem Dmentioning

confidence: 99%

String topology of finite groups of Lie type

Grodal,

Lahtinen

2020

Preprint

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“…Step 1: (The maximal torus normalizer and its automorphisms, [45,6]). A first step is to show that X and X ′ have isomorphic maximal torus normalizers.…”

Section: Uniqueness Of P-compact Groupsmentioning

confidence: 99%

“…A problem is however that N in general has too many automorphisms. To correct this, it was shown in [6] that the root subgroups N σ , introduced before Theorem 2.3, can also be built algebraically, and adding this extra data give the correct automorphisms. Concretely, one has a canonical factorization…”

Section: Uniqueness Of P-compact Groupsmentioning

confidence: 99%

“…For p-compact groups the lattices that come up are lattices over the p-adic integers Z p , rather than Z, so the concept of a root datum needs to be tweaked to make sense also in this setting, and one must carry out a corresponding classification. In this section we produce a short summary of this theory, based on [79,45,6,8]. In what follows R denotes a principal ideal domain of characteristic zero.…”

Section: Root Data Over the P-adic Integersmentioning

confidence: 99%

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The Classification of p-compact Groups and Homotopical Group Theory

Grodal

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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The classification of 2-compact groups

Andersen

Grodal

2008

J. Amer. Math. Soc.

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We prove that any connected 2 2 –compact group is classified by its 2 2 –adic root datum, and in particular the exotic 2 2 –compact group DI ⁡ ( 4 ) \operatorname {DI}(4) , constructed by Dwyer–Wilkerson, is the only simple 2 2 –compact group not arising as the 2 2 –completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for p p odd, this establishes the full classification of p p –compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p p –compact groups and root data over the p p –adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen–Grodal–Møller–Viruel methods by incorporating the theory of root data over the p p –adic integers, as developed by Dwyer–Wilkerson and the authors. Furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by Jackowski–McClure–Oliver in the early 1990s.

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Automorphisms ofp–compact groups and their root data

Cited by 3 publications

References 31 publications

String topology of finite groups of Lie type

String topology of finite groups of Lie type

The Classification of p-compact Groups and Homotopical Group Theory

The classification of 2-compact groups

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