We show that there exists a finitely generated group of growth for all functions satisfying for all large enough and the positive root of . Set ; then all functions that grow uniformly faster than are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth for a dense set of .
Hass, Rubinstein, and Scott showed that every closed aspherical (irreducible) 3-manifold whose fundamental group contains the fundamental group of a closed aspherical surface, is covered by Euclidean space. This theorem does not generalize to higher dimensions. However, we provide geometric tools with which variations of this theorem can be proved in all dimensions.
We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.
We provide a classification of isometries of systolic complexes corresponding to the classification of isometries of CAT(0)-spaces. We prove that any isometry of a systolic complex either fixes the barycentre of some simplex (elliptic case) or stabilizes a thick geodesic (hyperbolic case). This leads to an alternative proof of the fact that finitely generated abelian subgroups of systolic groups are undistorted.
Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define -Betti numbers and an -Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version of Poincaré...
On décrit le début du spectre des longueurs des groupes de triangles ayant un angle droit et on montre que le spectre des longueurs caractérise la classe d’isométrie d’un tel groupe.
We describe finitely generated groups universally equivalent (with constants from in the language) to a given torsion-free relatively hyperbolic group with free abelian parabolics. It turns out that, as in the free group case, the group embeds into the Lyndon’s completion of the group , or, equivalently, embeds into a group obtained from by finitely many extensions of centralizers. Conversely, every subgroup of containing is universally equivalent to . Since finitely generated...