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Sur la théorie élémentaire des groupes libres

Frédéric Paulin (2002/2003)

Séminaire Bourbaki

Sela a annoncé une solution complète d’un problème de Tarski, qui demanda vers 1945 quels sont les groupes de type fini qui ont la même théorie élémentaire qu’un groupe libre. Nous discuterons des travaux de Remeslennikov, Kharlampovich-Myasnikov, Sela, Champetier-Guirardel et autres sur la structure des groupes limites (les groupes de type fini qui sont “limites”de groupes libres, ou encore, qui ont la même théorie universelle qu’un groupe libre). Nous indiquerons quelques outils utilisés par Sela...

Sur l'accessibilité acylindrique des groupes de présentation finie

Thomas Delzant (1999)

Annales de l'institut Fourier

Soit G un groupe et τ un G -arbre. Dans cet article, nous supposons que G ne se scinde pas comme amalgame G = A * C B , ou HNN extension G = A * C au-dessus d’un groupe C qui stabilise un segment de longueur k dans τ ( k ≥ 2 ) ; si de plus τ ne contient pas de sous-arbre G -invariant, nous montrons que le nombre de sommets de τ / G est majoré par 12 k T , où T mesure la complexité d’une présentation de G .

Sur les treillis de Coxeter finis

C. Le Conte de Poly-Barbut (1994)

Mathématiques et Sciences Humaines

Björner (1984) a montré que l’ordre faible de Bruhat défini sur un groupe de Coxeter fini (Bourbaki 1969) est un treillis. Dans le cas du groupe symétrique S n ce résultat (treillis permutoèdre) a été prouvé par Guilbaud-Rosenstiehl (1963). Dans ce papier nous montrons que des propriétés connues des treillis permutoèdres peuvent s’étendre à tous les treillis de Coxeter finis et qu’inversement des propriétés démontrées sur tous les Coxeter finis ont des retombées intéressantes sur les permutoèdres....

Systolic groups acting on complexes with no flats are word-hyperbolic

Piotr Przytycki (2007)

Fundamenta Mathematicae

We prove that if a group acts properly and cocompactly on a systolic complex, in whose 1-skeleton there is no isometrically embedded copy of the 1-skeleton of an equilaterally triangulated Euclidean plane, then the group is word-hyperbolic. This was conjectured by D. T. Wise.

Testing Cayley graph densities

Goulnara N. Arzhantseva, Victor S. Guba, Martin Lustig, Jean-Philippe Préaux (2008)

Annales mathématiques Blaise Pascal

We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an m -generated group is amenable if and only if the density of the corresponding Cayley graph equals to 2 m . We test amenable and non-amenable...

The 4-string braid group B 4 has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

We prove that the braid group B 4 on 4 strings, its central quotient B 4 / 〈 z 〉 , and the automorphism group Aut ( F 2 ) of the free group F 2 on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group B 4 is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

The Bass conjecture and growth in groups

Anders Karlsson, Markus Neuhauser (2004)

Colloquium Mathematicae

We discuss Bass's conjecture on the vanishing of the Hattori-Stallings rank from the viewpoint of geometric group theory. It is noted that groups without u-elements satisfy this conjecture. This leads in particular to a simple proof of the conjecture in the case of groups of subexponential growth.

The Dehn functions of O u t ( F n ) and A u t ( F n )

Martin R. Bridson, Karen Vogtmann (2012)

Annales de l’institut Fourier

For n at least 3, the Dehn functions of O u t ( F n ) and A u t ( F n ) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n = 3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n bigger than 3 to the case n = 3 . In this note we give a shorter, more direct proof of this last reduction.

The geometry of abstract groups and their splittings.

Charles Terence Clegg Wall (2003)

Revista Matemática Complutense

A survey of splitting theorems for abstract groups and their applications. Topics covered include preliminaries, early results, Bass-Serre theory, the structure of G-trees, Serre's applications to SL2 and length functions. Stallings' theorem, results about accessibility and bounds for splittability. Duality groups and pairs; results of Eckmann and collaborators on PD2 groups. Relative ends, the JSJ theorems and the splitting results of Kropholler and Roller on PDn groups. Notions of quasi-isometry,...

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