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Presentations of finite simple groups: a computational approach

Robert Guralnick, William M. Kantor, Martin Kassabov, Alexander Lubotzky (2011)

Journal of the European Mathematical Society

All finite simple groups of Lie type of rank n over a field of size q , with the possible exception of the Ree groups 2 G 2 ( q ) , have presentations with at most 49 relations and bit-length O ( 𝚕𝚘𝚐 n + 𝚕𝚘𝚐 q ) . Moreover, A n and S n have presentations with 3 generators; 7 relations and bit-length O ( 𝚕𝚘𝚐 n ) , while 𝚂𝙻 ( n , q ) has a presentation with 6 generators, 25 relations and bit-length O ( 𝚕𝚘𝚐 n + 𝚕𝚘𝚐 q ) .

Resolutions of moduli spaces and homological stability

Oscar Randal-Williams (2016)

Journal of the European Mathematical Society

We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and gives explicit stability ranges in many new cases. In each of these cases the stable homology can be identified using the methods of Galatius, Madsen, Tillmann and Weiss.

Salvetti complex, spectral sequences and cohomology of Artin groups

Filippo Callegaro (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

Stable cohomology of alternating groups

Fedor Bogomolov, Christian Böhning (2014)

Open Mathematics

We determine the stable cohomology groups ( H S i 𝔄 n , 𝔄 n , p p of the alternating groups 𝔄 n for all integers n and i, and all odd primes p.

Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs.

Craig A. Jensen (2002)

Publicacions Matemàtiques

It is not known whether or not the stable rational cohomology groups H*(Aut(F∞);Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields...

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