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.     

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.     

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 rigidity for Seifert fibre spaces

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.     

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.     

An arithmetic property of profinite groups

We intend to generalize a crucial lemma of [4] to prove a somewhat surprising arithmetic property of profinite groups; namely, that a profinite group G has nontrivial p-Sylow-subgroups for only a finite number of primes if and only if this is true for its procyclic subgroups. This will yield as a corollary that every profinite torsion group has finite exponent if and only if this is true for its Sylow-sub-groups, a result also contained in [4].     

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     

Some open problems concerning relatively free and free profinite semigroups

Besides constituting an important and often crucial tool in studying semigroup varieties and pseudovarieties, relatively free and free profinite semigroups have proved to be of interest from many other natural standpoints. We present three groups of new problems concerning certain relatively free and free profinite semigroups and give background and motivation for them. The problems deal with the first-order theories of free profinite semigroups, the structure of free Burnside semigroups with commuting idempotents and of free Burnside inverse semigroups, and the representation of finitely generated free bands by order preserving transformations of a finite chain. April 19, 2001     

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.     

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.     

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.     

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.     

The probability of generating certain profinite groups by two elements

In this paper we consider the question of finite generation of profinite groups. We study the class of profinite groups which are inverse limits of wreath products of alternating groups of degree ≥5. We prove that the probability of generating such inverse limits by two elements is strictly positive and tends to 1 as the degree of the first factor tends to infinity. Our method of analysis requires a survey of the maximal subgroups of iterated wreath products of alternating groups. Although we have been unable to classify these precisely we do obtain upper bounds for the number of conjugacy classes of maximal subgroups which we believe to be of independent interest. The author is grateful for financial support received under the FCO-award scheme.     

Profinite群的共轭分离性和全形

本文讨论了profinite群的共轭分离性和全形，把关于无限群的结果推广到了profinite群。     

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.     

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.     

The Generalised Pro-Fitting Subgroup of a Profinite Group

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.