Let p be an odd prime number, k an imaginary abelian field containing a primitive p-th root of unity, and k∞/k the cyclotomic Zp-extension. Denote by L/k∞ the maximal unramified pro–p abelian extension, and by L′ the maximal intermediate field of L/k∞ in which all prime divisors of k∞ over p split completely. Let N/k∞ (resp. N ′/k∞) be the pro–p abelian extension generated by all p-power roots ...

Let Γ be a limit group, S ⊂ Γ a subgroup, and N the normaliser of S. If H1(S,Q) has finite Q-dimension, then S is finitely generated and either N/S is finite or N is abelian. This result has applications to the study of subdirect products of limit groups. Limit groups (or ω-residually free groups) have received a good deal of attention in recent years, primarily due to the groundbreaking work o...

Let G be a nite subgroup of GL d (Z). Then G acts on the Laurent polynomial ring kX 1 1 d ] over the eld k via the natural G-action on the multiplicative group generated by the variables X 1 ; : : : ; X d (= Z d). We show that the class group of the ring of invariants of this action is isomorphic to Hom(G=N;k) H 1 (G=D; (Z d) D), where N denotes the subgroup of G that is generated by all reeect...

We show that amalgams of nitely generated torsionfree nilpotent groups of class c along a cyclic subgroup satisfy a polynomial isoperimetric inequality of degree 4c. The distortion of the amalgamated subgroup is bounded above by a polynomial of degree c. We also give an example of a non-cyclic amalgam of nitely generated torsionfree nilpotent groups along an abelian, isolated and normal subgrou...

In this note, we prove that a random extension of either the free group FN of rank N ě 3 or of the fundamental group of a closed, orientable surface Sg of genus g ě 2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either OutpFN q or ModpSgq generated by k independent random walks. Our main theorem is that a k–generated random subgroup of ModpSgq or OutpFN ...

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of ...