1 → I g,r → M n g,r → Sp(2g,Z) → 1. We omit the decorations n and r when they are zero. In a series of papers [7, 8, 9, 10], D. Johnson obtained several fundamental results concerning the structure of Ig and Ig,1 (see also [11, 2]). In particular, he proved that Ig and Ig,1 are finitely generated for all g ≥ 3. On the contrary, A. Miller and D. McCullough [14] showed that I2 (and hence I n 2,r ...

The Milnor-Wolf Theorem characterizes the finitely generated solvable groups which have exponential growth; a finitely generated solvable group has exponential growth iff it is not virtually nilpotent. Wolf showed that a finitely generated nilpotent by finite group has polynomial growth; then extended this by proving that polycyclic groups which are not virtually nilpotent have expontial growth...

We consider Noether’s problem on the noncommutative rational functions invariant under a linear action of finite group. For abelian groups skew-fields are always rational, for solvable group they if is well-behaved – given by so-called complete representation. determine that admit such representations and call them totally pseudo-unramified. show skew-field finitely generated. Finally we study ...

In [ECHLPT] and [S] it is shown that if the fundamental group of a Seifert fibred 3manifold is not virtually nilpotent then it has an automatic structure. In the unpublished 1992 preprint [G2] Gersten constructs a biautomatic structure on the fundamental group of any circle bundle over a hyperbolic surface. He asks if the same can be done for the above Seifert fibered 3-manifold. We show the ex...

We show that, whenever Γ is a countable abelian group and ∆ ≤ Γ is a finitely-generated subgroup, a generic measure-preserving action of ∆ on a standard atomless probability space (X,μ) extends to a free measure-preserving action of Γ on (X,μ). This extends a result of Ageev, corresponding to the case when ∆ is infinite cyclic.

We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with constant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously, and in the case of finitely generated Abelian groups the necessary and sufficient conditio...

It is shown, for a given graph group G, that the fixed point subgroup Fix' is finitely generated for every endomorphism ' of G if and only if G is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms if the dependence graph of G is a transitive forest.

This self-contained paper is part of a series [FF1, FF2] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante-Thurston proved that every nilpotent subgroup of Diff(S) is abelian. One of our main results is a sharp converse: Diff(S) contains every finitely-generated, torsion-free nilpotent group.

Let Qp ⊆ L ⊆ K ⊆ Cp be a chain of complete intermediate fields where Qp ⊆ L is finite and K discretely valued. Let Z be a one dimensional finitely generated abelian locally L-analytic group and let ẐK be its rigid Kanalytic character group. Generalizing work of Lazard we compute the Picard group and the Grothendieck group of ẐK . If Z = o, the integers in L 6= Qp, we find Pic(ôK) = Zp which ans...

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only on S.