The Asymptotic Density of Finite-order Elements in Virtually Nilpotent Groups

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

N ov 2 00 4 Bounded geometry in relatively hyperbolic groups

We prove that a group is hyperbolic relative to virtually nilpotent subgroups if and only if there exists a Gromov-hyperbolic metric space with bounded geometry on which it acts as a relatively hyperbolic group. As a consequence we obtain that any group hyperbolic relative to virtually nilpotent subgroups has finite asymptotic dimension. For these groups the Novikov conjecture holds. The class ...

Fine Asymptotic Geometry in the Heisenberg Group

For every finite generating set on the integer Heisenberg group H(Z), we know from a fundamental result of Pansu on nilpotent groups that the wordmetric has the large-scale structure of a Carnot-Carathéodory Finsler metric on the real Heisenberg group H(R). We study the properties of those limit metrics and obtain results about the geometry of word metrics that reflect the dependence on generat...

Cohomological Finiteness Conditions in Bredon Cohomology

We show that soluble groups G of type Bredon-FP∞ with respect to the family of all virtually cyclic subgroups of G are always virtually cyclic. In such a group centralizers of elements are of type FP∞. We show that this implies the group is polycyclic. Another important ingredient of the proof is that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic ...

ZETA FUNCTIONS AND COUNTING FINITE p-GROUPS

We announce proofs of a number of theorems concerning finite p-groups and nilpotent groups. These include: (1) the number of p-groups of class c on d generators of order pn satisfies a linear recurrence relation in n; (2) for fixed n the number of p-groups of order pn as one varies p is given by counting points on certain varieties mod p; (3) an asymptotic formula for the number of finite nilpo...