Buildings

Abramenko, Peter; Brown, Kenneth S.

doi:10.1007/978-0-387-78835-7

Cited by 302 publications

(425 citation statements)

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“…The books by Abramenko and Brown [AB08] and by Ronan [Ron89] are excellent references, where all of the material of this section can be found.…”

Section: Spherical Buildingsmentioning

confidence: 99%

Homological stability for classical groups

Essert¹

2013

Isr. J. Math.

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Associated to every group with a weak spherical Tits system of rank n + 1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields. IntroductionHomological stability is the following question: Given an infinite series of groups G n , such as the general linear groups GL n , we consider the sequence of inclusionsThen, if we apply group homology of a fixed degree, does the corresponding sequence of homology modules stabilise eventually? This is an old question and there are many interesting results for various series of classical groups, usually over rings of finite stable rank. An overview of results in this area can be found in [Knu01, Chapter 2] and we will also provide references to the best known results for specific series of groups. Although the method of proof is usually based on a common idea, the action of the larger group on a highly connected simplicial complex, all proofs known to the author are tailored to specific series of groups.In this paper, we present a general method to prove homological stability, valid for all groups with weak spherical Tits systems, that is, groups acting strongly transitively on possibly weak spherical buildings. We then use this method to prove homological stability for various series of classical groups over division rings, usually improving the stability range previously known from work with larger classes of rings.The method is based on the observation that the simplicial complexes used by Charney in [Cha80]

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“…The books by Abramenko and Brown [AB08] and by Ronan [Ron89] are excellent references, where all of the material of this section can be found.…”

Section: Spherical Buildingsmentioning

confidence: 99%

Homological stability for classical groups

Essert¹

2013

Isr. J. Math.

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“…This poset is an abstract simplicial complex as defined in section 1.1; moreover, it is a thin chamber complex [3,Theorem 3.5].…”

Section: Coxeter Complexesmentioning

confidence: 99%

“…Definition (4.1 in [3]). A building is a simplicial complex ∆ that can be expressed as the union of subcomplexes Σ (called apartments) satisfying the following axioms:…”

Section: Buildingsmentioning

confidence: 99%

“…On the other hand, given a set C and a function δ : C × C → W satisfying these properties, there exists a building, unique up to isomorphism, such that C = C(∆) with δ as its Weyl distance function [3,Corollary 5.93]. The simplices in this building correspond to the residues of C, which are certain distinguished sets of chambers; for a precise description see [3, §5.6].…”

Section: The δ Function and Twin Buildingsmentioning

confidence: 99%

“…Definition (6.55 in [3]). Subgroups B and N of a group G form a BN-pair if they generate G, the intersection T B ∩ N is normal in N, and the quotient W N/T (the Weyl group) admits a set of generators S such that the following hold:…”

Section: Buildings and Groupsmentioning

confidence: 99%

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