Profinite completions of some groups acting on trees

We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups.     

Grothendieck''s problem on profinite completions and representations of groups

The paper is concerned with Grothendieck's problem on profinite completions of groups. The relationship of this problem to the representation theory of finitely generated groups and to the problem of arithmeticity of Platonov are treated.To Professor A. Grothendieck on the Occasion of his 60th Birthday     

On profinite completions and canonical extensions

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V. The converse holds for varieties of finite type. This paper is dedicated to Walter Taylor. Received May 14, 2005; accepted in final form September 8, 2005.     

Natural extensions and profinite completions of algebras

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class \({\mathcal{A} = \mathbb{ISP}(\mathcal{M})}\), where \({\mathcal{M}}\) is a set, not necessarily finite, of finite algebras, it is shown that each \({{\bf A} \in \mathcal{A}}\) embeds as a topologically dense subalgebra of a topological algebra \({n_{\mathcal{A}}({\bf A})}\) (its natural extension), and that \({n_{\mathcal{A}}({\bf A})}\) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that \({\mathcal{M}}\) is finite and \({\mathcal{A}}\) possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.     

A note on profinite completions and canonical extensions

J. Harding has proved that the profinite limit of an algebra A in a finitely generated variety of monotone lattice expansions coincides with its canonical extension. In this note we drop the monotonicity of the additional operations and prove the same result.     

Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

We show that the class of pairs (Γ,H) of a group and a finite index subgroup which verify a conjecture of Moore about projectivity of modules over ZΓ satisfy certain closure properties. We use this, together with a result of Benson and Goodearl, in order to prove that Moore's conjecture is valid for groups which belongs to Kropholler's hierarchy LHF. For finite groups, Moore's conjecture is a consequence of a theorem of Serre, about the vanishing of a certain product in the cohomology ring (the Bockstein elements). Using our result, we construct examples of pairs (Γ,H) which satisfy the conjecture without satisfying the analog of Serre's theorem.     

Profinite groups of finite cohomological dimension

Let 1→N→G→G/N→1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n:=cd_p(G)<∞ and if the order of $\text{H}{}^{\mathrm{k}}\text{(N,}\text{F}{}_{\mathrm{p}}\text{)}$ is finite for k:=cd_p(N), the virtual cohomological p-dimension of G/N equals n?k. To cite this article: T. Weigel, P. Zalesskii, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Simple groups,maximal subgroups,and probabilistic aspects of profinite groups

We show that a finite simple group has at mostn ^1.875+o(1) maximal subgroups of indexn. This enables us to characterise profinite groups which are generated with positive probability by boundedly many random elements. It turns out that these groups are exactly those having polynomial maximal subgroup growth. Related results are also established.     

New results related to a conjecture of Moore

Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore’s conjecture: Assume that, for all ${x \in (\Gamma-\Gamma^{\prime})}$ , either there is an integer n such that ${1 \neq x^{n} \in \Gamma^{\prime}}$ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler’s hierarchy ${{\bf LH}\mathfrak{F}}$ . In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index.     

Permutation modules of profinite groups

Given an arbitrary profinite group G and a commutative domain R, we define the notion of permutation RG-module which generalizes the known notion from the representation theory of profinite groups. We establish an independence theorem of such a module as an R-module over a ring of scalars.     

Isometry types of profinite groups

Let T be a rooted tree and Iso(T) be the group of its isometries. We study closed subgroups G of Iso(T) with respect to the number of conjugacy classes of Iso(T) having representatives in G.     

Modular representations of profinite groups

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra of a profinite group G, where k is a finite field of characteristic p.We define the concept of relative projectivity for a profinite -module. We prove a characterization of finitely generated relatively projective modules analogous to the finite case with additions of interest to the profinite theory. We introduce vertices and sources for indecomposable finitely generated -modules and show that the expected conjugacy properties hold—for sources this requires additional assumptions. Finally we prove a direct analogue of Green’s Indecomposability Theorem for finitely generated modules over a virtually pro-p group.     

Hyperbolic spaces at large primes and a conjecture of Moore

A simply connected finite complex X is called elliptic if its rational homotopy Lie algebra is of finite dimension and hyperbolic otherwise. According to a conjecture of Moore, there exists an exponent for the p-torsion part of if and only if X is elliptic. In this note, it is shown that, provided the prime p is sufficiently large, a hyperbolic space with p-torsion free loop space homology has no exponent in the p-torsion of the homotopy groups. For a class of formal spaces, this result is obtained for every odd prime.     

Cohomology and finite subgroups of profinite groups

We prove two theorems linking the cohomology of a pro- group with the conjugacy classes of its finite subgroups.

The number of conjugacy classes of elementary abelian -subgroups of is finite if and only if the ring is finitely generated modulo nilpotent elements.

If the ring is finitely generated, then the number of conjugacy classes of finite subgroups of is finite.