Profinite completions and isomorphic finite quotients

Čech systems that induce approximate inverse systems

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

On groups with uncountably many subgroups of finite index

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

Generalizing results in [1], [3], [5], and [7] it is shown that the algebra generated by the quasi-regular representation λ of a discrete groupG onl2 (G/H) satisfies a standard polynomial identity if and only if λ (G) contains an abelian normal subgroup of finite index. An analogon for topological groups is also proved.

We shall often need to consider the element Z =X Y, which is a parabolic transformation. Any parabolic transformation in r is conjugate to a power of Z. Let N be a normal subgroup of finite index ,u in r. The level n of N is defined as the least positive integer such that ZnCN. The conjugacy class of Z in r splits up into a finite number of equivalence classes under conjugacy by N. The number t of equivalence classes is called the parabolic class number of N. It is known that the integers ,u, n, t satisfy the relation

Let \(\mathcal{N}\) be a nonempty collection of normal subgroups of finite index of a group G and assume that \(\mathcal{N}\) is filtered from below, i.e., \(\mathcal{N}\) satisfies the following condition: $$\mbox{whenever }N_1,N_2\in \mathcal{N},\mbox{ there exists }N\in \mathcal{N}\mbox{ such that }N\le N_1\cap N_2.$$ Then one can make G into a topological group by considering \(\mathcal{N}\) as a fundamental system of neighborhoods of the identity element 1 of G. We refer to the corresponding topology on G as a profinite topology. If every quotient G/N\((N\in \mathcal{N})\) belongs to a certain class \(\mathcal{C}\), we say more specifically that the topology above is a pro - \(\mathcal{C}\)topology.

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.

The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows.

Let ~ be a finite set (called symbols) with the discrete topology and ~ = ~9~ be the set of all doubly infinite sequences of symbols with the product topology. We shall be interested in the homeomorphism ~:Zg~+ E~ which shifts a sequence one space to the left. In symbolic dynamics one studies a differentiable system (diffeomorphism or flow) by relating it to the shift homeomorphism ~. This idea started with Hadamard [ii] in 1898 when he set up a correspondence between geodesics on a surface of negative curvature with certain symbolic sequences. Then Marston Morse used this correspondence to construct a nonperiodic almost periodic geodesic [15], [16], [17]. Over the years Professor Hedlund has been active in symbolic dynamics (see for example [12] and [13]).

A resolution for the product of a compactum with a polyhedron

In this paper the results of [5] are extended to classes of lattices. We assume familiarity with [5], but we recall for convenience the principal definitions and notations. If is a category and if is a direct [resp., inverse] limit system in , then is the direct [resp., inverse] limit of (determined only up to isomorphism in ). If is an inverse limit system of sets or universal algebras, let denote the canonical construction of inverse limit described for example in [1, Chapter 3].

Constructing quantized enveloping algebras via inverse limits of finite dimensional algebras

We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$.

Decision problems and profinite completions of groups