Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property

The focus of this chapter is on sequences of kleinian groups, typically sequences that are becoming degenerate in some way. For these, it is necessary to distinguish between convergence of groups and convergence of quotient manifolds. The former has to do with sequences of groups whose generators converge, the latter with sequences of groups whose fundamental polyhedra converge. We will continue on by introducing the operations called Dehn filling and surgery. This leads to the description of the set of volumes of finite volume hyperbolic 3-manifolds. Algebraic convergence In this section we will prove the two theorems which provide the basis for working with sequences of groups. Let Γ be an abstract group and { φ n : Γ → G n } be a sequence of homomorphisms (also called representations ) { φ n } of Γ to groups G n of Mobius transformations. Suppose for each γ ∈ Γ, lim n →∞ φ n (γ) = φ(γ) exists as a Mobius transformation. Then the sequence { φ n } is said to converge algebraically and its algebraic limit is the group of limits G ∞ = { φ(γ) : γ ∈ Γ}. When we say a sequence of groups converges algebraically, we are assuming that behind the statement is a sequence of homomorphisms or isomorphisms spawning the sequence. In particular, a sequence of r -generator groups G n = ⟨ A 1 , n , A 2 , n , … A r,n ⟩ is said to converge algebraically if A k = lim n →∞ A k ,n exists as a Mobius transformation, 1 ≤ k ≤ r . Its algebraic limit is the group G = ⟨ A 1 , A 2 , … A r ⟩. To make this terminology consistent with that used above, refer to the free group F r on r -generators and express G n as the sequence of representations φ n : F r → G n determined by sending the k -th generator of F r to A k,n . The convergence is controlled as spelled out by the following two fundamental results. Theorem 4.1.1 [Jorgensen 1976; Jorgensen and Klein 1982]. Let { G n } be a sequence of r-generator nonelementary kleinian groups converging algebraically to the group G. Then G is also a nonelementary kleinian group, and the map A k → A k,n , 1 ≤ k ≤ r, determines a homomorphism ϕ n : G → Gn for all large indices n . In general ϕ n will not be an isomorphism.

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.

Let G be a finite group. We let [Formula: see text] and σ (G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/ Φ (G) when G is a finite soluble group with [Formula: see text].

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio [Formula: see text]. This extends the case [Formula: see text] studied in previous papers [R. Brandl and W. Shi, Finite groups whose element orders are consecutive integers, J. Algebra 143 (1991) 388–400; Y. Feng, Finite groups whose abelian subgroup orders are consecutive integers, J. Math. Res. Exp. 18 (1998) 503–506; W. Shi, Finite groups whose proper subgroup orders are consecutive integers, J. Math. Res. Exp. 14 (1994) 165–166].

Abstract. We obtain a branched spherical CR structure on the com- plement of the figure-eight knot whose holonomy representation was given in [4]. There are essentially two boundary unipotent representations from the complement of the figure-eight knot into PU(2,1), which we call ρ1 and ρ2. We make explicit some fundamental differences between these two representations. For instance, seeing the figure-eight knot complement as a surface bundle over the circle, the behavior of the fundamental group of the fiber under the representa- tion is a key difference between ρ1 and ρ2.

Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G is only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M.

A finite group G is said to be a minimal non-[Formula: see text] group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of p-groups H of odd order with d(H) = 2 and c(H) = 2. Based on the classification of H, minimal non-[Formula: see text]p-groups G are classified for p > 3. If p > 3, then we have G3 ≅ Cp or G3 ≅ Cp × Cp. In this paper, we deal with the case when G3 ≅ Cp. In another paper [A classification of finite p-groups whose proper subgroups are of class ≤ 2(II), accepted] we deal with the case when G3 ≅ Cp × Cp.

Let G be a finite group. If there is a set containing n proper subgroups in G whose union comprises all elements of G, then G is a coverable group. The set of n proper subgroups is called an n-covering for G and the minimum number of n covers G is 3, so this means. If the n-covering has no proper subset that also covers G then it is called an irredundant n-covering for G and it is called a maximal n-covering if it consists of only maximal subgroups. A maximal irredundant n-covering with a core-free intersection is known as a Cn-covering. This study characterizes 5-groups having a C11-covering. It was found that a 5-group has a C11-covering if and only if it is isomorphic to some elementary abelian groups of a certain order.

A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup \(A\) of a group \(G\) is semi-normal in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=AB\) and \(AB_1\) is a proper subgroup of \(G\) for every proper subgroup \(B_1\) of \(B\). If \(A\) is either subnormal in \(G\) or is semi-normal in \(G\), then \(A\) is called a semi-subnormal subgroup of \(G\). In this paper, we establish that a group \(G\) with semi-subnormal Schmidt \(\{2,3\}\)-subgroups is \(3\)-soluble. Moreover, if all 5-closed Schmidt \(\{2,5 \}\)-subgroups are semi-subnormal in \(G\), then \(G\) is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent.

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.

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 $\psi\in{\rm{Out}}(F_r)$.

A group is called metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite locally graded group whose proper subgroups are metahamiltonian is likewise metahamiltonian, and the aim of this paper is to describe the structure of locally graded groups whose proper subgroups contain a metahamiltonian subgroup of finite index.

In this paper we study normal subgroups of Kleinian groups as well as discrepancy groups (d-groups), that are Kleinian groups for which the exponent of convergence is strictly less than the Hausdorff dimension of the limit set. We show that the limit set of a d-group always contains a range of fractal subsets, each containing the set of radial limit points and having Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. We then consider normal subgroups $G$ of an arbitrary non-elementary Kleinian group $H$, and show that the exponent of convergence of $G$ is bounded from below by half of the exponent of convergene of $H$. Finally, we give a discussion of various examples of d-groups.

April, 1992 Abstract. We study a class of centrally trivial automorphisms for subfactors, and get an upper bound for the order of the group they make (modulo normalizers) in terms of the “dual” principal graph for AFD type II1 subfactors with trivial relative commutant, finite index and finite depth. We prove that this upper bound is attained for many known subfators. We also introduce χ(M,N) for subfactors N ⊂ M as the relative version of Connes’ invariant χ(M), and compute this group for many AFD type II1 subfactors with finite index and finite depth including all the cases with index less than 4 and many Hecke algebra subfactors of Wenzl. In these finite depth cases, the group χ(M,N) is always finite and abelian, and we realize all the finite abelian groups as χ(M,N). Analogy between this topic and modular structure of type III factors is also discussed. As an application, we give some classification results for Aut(M,N). For example, for the subfactors of type A2n+1, there are two and only two outer actions of Z2. One is of the “standard” form and the other is given by the “orbifold” action arising from the paragroup symmetry. As preliminaries, we also prove several statements on central sequence subfactors announced by A. Ocneanu.