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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,...

The rate of escape for random walks on polycyclic and metabelian groups

Russ Thompson (2013)

Annales de l'I.H.P. Probabilités et statistiques

We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on some abelian-by-cyclic groups via a comparison to the toppling of a dissipative abelian...

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