Continuous Quotients for Lattice Actions on Compact Spaces

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

We show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in Nickel. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented by Nickel.

In this last chapter we want to give some idea of the “anabelian” program of A. Grothendieck. The term anabelian should be read as “far from being abelian” and as we understand the matter, a group is far enough away from being abelian if all of its subgroups of finite index have a trivial center. The principal idea is the following: in topology, a space X of type K(π,1) is determined by its fundamental group π up to weak homotopy equivalence. If we require that X is a CW-complex, then X is already determined up to strong homotopy equivalence. The “anabelian” idea is that something similar should also be true for schemes, i.e. a scheme X which is an étale K(π,1) should be essentially reconstructible from its étale fundamental group. This is obviously not correct in general, but it should be true under certain conditions; for example, X should be absolutely finitely generated and $\pi^{\mathit{et}}_{1}(X)$ is supposed to be “anabelian”. The smallest constituents of this anabelian world are points, i.e. spectra of fields which are finitely generated over their prime fields. Here the étale fundamental group is just the absolute Galois group. Finite fields have an abelian absolute Galois group (in fact all these fields have the same Galois group $\hat{\mathbb{Z}}$ ), and so the first objects of interest are global fields. In §1,2 we will present some results on “anabelian properties” of global fields, which already existed before Grothendieck formulated his program. We will explain the general conjectures in §3.

Algorithms for Matrix Groups and the Tits Alternative

Hyperbolic 3-manifolds whose fundamental groups are subgroups of finite index in right-angled Coxeter groups are under consideration. The construction of such manifolds is associated with regular colourings of the faces of polyhedra and, in particular, with 4-colourings. The following questions are discussed: the structure of the set of right-angled polytopes in Lobachevskii space; examples of orientable and non-orientable manifolds, including the classical Löbell manifold constructed in 1931; connections between the Hamiltonian property of a polyhedron and the existence of hyperelliptic involutions of manifolds; the volumes and complexity of manifolds; isometry between hyperbolic manifolds constructed from 4-colourings. Bibliography: 89 titles.

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of boundedly generated groups (examples that come close to achieving these bounds are presented). This proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov that asserts that the rank gradient of infinite boundedly generated residually finite groups is zero. Furthermore, our first result establishes a variant of a conjecture of Lubotzky on the ranks of finite index subgroups of special linear groups over the integers, and is analogous to a result of Pyber and Segal for solvable groups.

Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.

On groups with uncountably many subgroups of finite index

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Given a free Abelian group of finite rank, canonical subgroup matrices are defined with respect to an arbitrary but fixed basis of , which uniquely determine all subgroups of of finite index. The use of these matrices allows the classification of all subgroups of fixed finite index. An application is made in the construction of the intersection group and the union group of two arbitrary subgroups of finite index and upper and lower bounds on the index of these groups are found. Physical applications occur for rank three in the systematic symmetry analysis of domain structures, which appear in structural phase transitions in solids.

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.

In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections. In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.

A profinite group is called strongly complete if every subgroup of finite index is open and of type (AF) if it has only finitely many subgroups of any fixed index. In this paper it is shown that a topologically finitely generated abelian by pro-nilpotent profinite group is strongly complete, and that a pro-solvable profinite group is strongly complete if is of type (AF).