Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP_n over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FP_n is closed under extensions, quotients by subgroups of type FP_n, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP_∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP_n but not of type FP_n+1 over Z_? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP_n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.     

Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-module, whose restriction to RH is projective.Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ, at least one of the following two conditions holds:

(M1)

〈x〉∩H≠{e} (in particular this holds if Γ is torsion free)

(M2)

ord(x) is finite and invertible in R.

Then M is projective as an RΓ-module.More generally, the conjecture has been formulated for crossed products R*Γ and even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free.The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.     

Presentations of Schützenberger groups of minimal subshifts

In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J -classes of free profinite semigroups. In this paper, the Schützenberger groups of such J -classes are investigated, in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for all non-periodic minimal subshifts associated with substitutions. It entails that it is decidable whether a finite group is a quotient of such a profinite group. As a further application, the Schützenberger group of the J -class corresponding to the Prouhet-Thue-Morse subshift is shown to admit a somewhat simpler presentation, from which it follows that it has rank three, and that it is non-free relatively to any pseudovariety of groups.     

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the fibers are of unbounded size.     

Permanence criteria for semi-free profinite groups

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x, y)) have free absolute Galois groups.     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.     

Relative cohomology theory for profinite groups

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer–Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components.     

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.     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

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.     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.     

Geometric stability theory for μ-structures

We introduce a notion of μ-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures (introduced by Krupinski in [9]). Using the Haar measure of locally compact groups, we introduce an independence, called μ-independence, in μ-structures having good properties. With this independence notion, we develop geometric stability theory for μ-structures. Then we see some structural theorems for compact groups which are μ-structure. We also give examples of profinite structures where μ-independence is different from nm-independence introduced by Krupinski for Polish structures.In an appendix, Cohen and Wesolek show that a profinite branch group gives a small action on the boundary of a rooted tree so that this actions provides a small profinite structure on the boundary of a rooted tree.     

Homological properties of abstract and profinite modules and groups

Generalising the main results from [F. Grunewald, A. Jaikin, A. Pinto, P. Zalesskii, Normal subgroups of profinite groups of non-negative deficiency (preprint); D.H. Kochloukova, On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version, J. Pure Appl. Algebra 204 (3) (2006) 536-554; D.H. Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups, Comment. Math. Helv. 81 (4) (2006) 931-943] we study homological finiteness properties and projective dimensions of modules over groups G in the abstract and profinite case comparing with the same invariants for some specific normal subgroups N of G.     

On groups whose subgroups are closed in the profinite topology

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.     

Continuous group actions on profinite spaces

For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point spaces and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite étale topological types of varieties over a field. One motivation is to understand Grothendieck’s section conjecture in terms of homotopy fixed points.     

Positively finitely related profinite groups

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.     

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.     

Procyclic coverings of commutators in profinite groups

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G′ such that the order of M is m-bounded and G′/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G′ is either finite of m-bounded order or procyclic.     

Small profinite structures

We propose a model-theoretic framework for investigating profinite structures. We prove that in many cases small profinite structures interpret infinite groups. This corresponds to results of Hrushovski and Peterzil on interpreting groups in locally modular stable and o-minimal structures.

     

Profinite Groups Admitting Just Infinite Quotients

 A profinite group is said to be just infinite if each of its proper quotients is finite. We address the question which profinite groups admit just infinite quotients. It is proved that any profinite group whose order (as a supernatural number) is divisible only by finitely many primes admits just infinite quotients. It is shown that if a profinite group G possesses the property in question then so does every open subgroup and every finite extension of G. Received 20 July 2001