Cohomology Rings of Almost-direct Products of Free Groups
نویسنده
چکیده
An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space. 1. Almost direct products of free groups If G1 and G2 are groups, and α : G1 → Aut(G2) is a homomorphism from G1 to the group of (right) automorphisms of G2, the semidirect product G = G2 ⋊α G1 is the set G2 ×G1 with group operation (g2, g1) · (g ′ 2, g ′ 1) = (α(g ′ 1)(g2)g ′ 2, g1g ′ 1). There is a corresponding split, short exact sequence 1 // G2 ι2 // G π // G1 // ι1 vv 1, where ι1(g1) = (1, g1), ι2(g2) = (g2, 1), and π(g2, g1) = g1. Identifying G1 and G2 with their images under ι1 and ι2, the group G is generated by G1 and G2. Furthermore, for g1 ∈ G1 and g2 ∈ G2, the relation g −1 1 g2g1 = α(g1)(g2) holds in G. If G1 and G2 are free groups, these are the only relations in G. An almost-direct product of free groups is an iterated semidirect product G = ⋊i=1Fni = Fnl ⋊αl (Fnl−1 ⋊αl−1 (· · ·⋊α3 (Fn2 ⋊α2 Fn1))) of finitely generated free groups in which the action of the group ⋊ji=1Fni onH1(Fnk ;Z) is trivial for each j and k, 1 ≤ j < k ≤ l. In other words, the automorphisms αk : ⋊ k−1 i=1 Fni → Aut(Fnk) which determine the iterated semidirect product structure of G are IA-automorphisms, inducing the identity on the abelianization of Fnk . If Fni is freely generated by xi,p, 1 ≤ p ≤ ni, the group G is generated by these elements (for 1 ≤ i ≤ l), and has defining relations (1.1) x i,pxj,qxi,p = αj(xi,p)(xj,q), 1 ≤ i < j ≤ l, 1 ≤ p ≤ ni, 1 ≤ q ≤ nj. 2000 Mathematics Subject Classification. 20F28,20F36,20J06,55M30.
منابع مشابه
Ring structures of mod p equivariant cohomology rings and ring homomorphisms between them
In this paper, we consider a class of connected oriented (with respect to Z/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = Z/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes ...
متن کاملLocal Cohomology of Segre Product Type Rings
The aim of this paper is to investigate properties of the local cohomology of rings of mixed characteristic that are analogous to Segre products of rings de ned over a eld. The main question is whether the local cohomology can be almost killed in a nite extension (we de ne what this means below). There are two reasons for considering this type of ring. First, there are special properties of the...
متن کاملRings for which every simple module is almost injective
We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...
متن کاملUniversal rings arising in geometry and group theory
Various algebraic structures in geometry and group theory have appeared to be governed by certain universal rings. Examples include: the cohomology rings of Hilbert schemes of points on projective surfaces and quasiprojective surfaces; the Chen-Ruan orbifold cohomology rings of the symmetric products; the class algebras of wreath products, as well as their associated graded algebras with respec...
متن کاملIntegrality of L2-Betti numbers
The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that H G is a normal subgroup, for amalgamated free products G ∗H (H ⋊ F ). Here F is a free group and H ⋊ F is an arbitrary semi-direct product. This includes free products G∗F and semi-...
متن کامل