Right-angled polyhedra and hyperbolic 3-manifolds

We consider limits of inverse sequences of closed manifolds, whose consecutive terms are obtained by connect summing with closed manifolds, which are in turn trivialized by the bonding maps. Such spaces, which we refer to as trees of manifolds, need not be semilocally simply connected at any point and can have complicated fundamental groups. Trees of manifolds occur naturally as visual boundaries of standard nonpositively curved geodesic spaces, which are acted upon by right-angled Coxeter groups whose nerves are closed PL-manifolds. This includes, for example, those Coxeter groups that act on Davis' exotic open contractible manifolds. Also, all of the homogeneous cohomology manifolds constructed by Jakobsche are trees of manifolds. In fact, trees of manifolds of this type, when constructed from PL-homology spheres of common dimension at least 4, are boundaries of negatively curved geodesic spaces. We prove that if Z is a tree of manifolds, the natural homomorphism φ: π 1 (Z) → π 1 (Z) from its fundamental group to its first shape homotopy group is injective. If Z = bdy X is the visual boundary of a nonpositively curved geodesic space X, or more generally, if Z is a Z-set boundary of any ANR X, then the first shape homotopy group of Z coincides with the fundamental group at infinity of X: π 1 (Z) = π 1 ∞ (X). We therefore obtain an injective homomorphism ψ: π 1 (Z) → π 1 ∞ (X), which allows us to study the relationship between these groups. In particular, if Z = bdy r is the boundary of one of the Coxeter groups Γ mentioned above, we get an injective homomorphism ψ: π 1 (bdy F) → π 1 ∞ (Γ).

Motivated by the relation between the Chern–Simons gauge theory and (2+1)-dimensional gravity, we find a formulation of gauge theories which applies to both orientable and non-orientable manifolds, using orientation bundles and density-valued forms. We show that on a non-orientable manifold, (2+1)D gravity is equivalent to BF theory, which is still topological and can be mapped in turn to Chern–Simons theory on the orientable double cover. By quantizing U(1) BF theory on a non-orientable manifold, we find that non-orientability introduces additional constraints on the quantized BF theory beyond those present for an orientable manifold, such that the coupling constant can only adopt a small number of discrete values. Specifically, for both the Klein bottle of demigenus 2 (N2) and the compact surface of non-orientable genus 3 (Dyck’s surface or N3), we find explicit representations for the holonomy, large gauge, and mapping class groups, as well as the Hilbert space; here the above symmetries along with the non-orientability of the surface constrain the coupling constant k to only take values 1/2, 1, or 2.

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.

We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are [Formula: see text]-free for a given integer [Formula: see text]. We show that any such manifold [Formula: see text] contains a point [Formula: see text] with the following property: If [Formula: see text] is the set of maximal cyclic subgroups of [Formula: see text] that contain non-trivial elements represented by loops of [Formula: see text], then for every subset [Formula: see text], we have rank [Formula: see text]. This generalizes to all [Formula: see text] results proved in [J. W. Anderson, R. D. Canary, M. Culler and P. B. Shalen, Free Kleinian groups and volumes of hyperbolic 3-manifolds, J. Differential Geom. 43 (1996) 738–782; M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156] for [Formula: see text]. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136; R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156].

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 consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. We provide some first applications. In addition, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover a result stating that if the defining graph contains no SILs, then Aut^0(W) is a right-angled Coxeter group.

Let Γ < SL n ( $$\mathbb{Z}$$ ) be a subgroup of finite index, where n ≥ 5. Suppose Γ acts continuously on a manifold M, where π1(M) = $$\mathbb{Z}$$ n , preserving a measure that is positive on open sets. Further assume that the induced Γ action on H 1(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ : M → $$\mathbb{T}$$ n that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where π1(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with Γ actions.

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group’s defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group’s order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n &gt; 1 , we give examples of right-angled Coxeter groups which are thick of order n , yet are algebraically thick of order strictly larger than n, answering a question of Behrstock-Druţu-Mosher.

Algorithms for Matrix Groups and the Tits Alternative

The main theme of this paper is higher virtual algebraic fibering properties of right‐angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type but not , and examples where the RACG is hyperbolic and the kernel is finitely generated and non‐hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz–Norin–Wise involving Bestvina–Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

By classical results, a compact Riemann surface, or equivalently an orientable closed hyperbolic 2-manifold, has at most one hyperelliptic involution (i.e. the quotient by the involution is the 2-sphere). A hyperelliptic involution is central in the automorphism or isometry group of such a hyperelliptic surface, its fixed points are exactly the Weierstrass points of the surface (see e.g. [FK]). In the present paper, we consider hyperelliptic involutions of hyperbolic 3manifolds; by definition, these are isometric involutions whose quotient is the 3-sphere. The situation is more complicated in dimension three, in particular a hyperbolic 3-manifold can have more than one hyperelliptic involution. Our main result states that there is a universal bound on the number of non-conjugate hyperelliptic involutions of closed hyperbolic 3-manifolds; this bound is quite small, and we compute it for various situations. Our methods also determine how different hyperelliptic involutions are related. There remains a small gap between these upper bounds and the number of different involutionswhichwe can actually construct (see [R1],[RZ]).We remark that the question on the number of different hyperelliptic involutions of hyperbolic 3-manifolds has explicitly been asked in [MV]. We obtain also results for hyperelliptic isometries of orders greater than two; in particular, for prime numbers p > 2, we give a complete solution to the problem on the number of different hyperelliptic isometries of order p (i.e. the upper bound we obtain is sharp). Let Isom+(M) denote the orientation preserving isometry group of a hyperbolic 3-manifoldM .

It was shown by Chinburg and Reid that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert showed that almost all quasi-Fuchsian 3-manifolds have all closed geodesics simple and disjoint. The natural conjecture arose that the Chinburg-Reid examples also had disjoint geodesics. Here we show that this conjecture is both almost true (they have no geodesics that intersect except at right angles) and spectacularly false (any pair of closed geodesics admits infinitely many closed geodesics which intersects both geodesics of the pair perpendicularly). The latter statement is shown to be true for all closed arithmetic hyperbolic 3-manifolds. Section 0 Introduction By a hyperbolic n-manifold we shall mean a complete orientable n-dimensional Riemannian manifold all of whose sectional curvatures are −1. If M is a hyperbolic 3-manifold, the universal cover of M can be identified with H, the upper half-space model of hyperbolic 3-space, andM is realized asH/Γ for some Γ a discrete torsionfree subgroup of Isom(H3). Now Isom(H3) can be identified with PSL(2,C) ( which in turn is isomorphic to PGL(2,C)), and Γ is called a Kleinian group. In the sequel we will only be interested in the case when M is closed, in which case Γ is referred to as cocompact. Of interest to us is the structure of the set of closed geodesics in closed hyperbolic 3-manifolds. The motivation comes from the following. A closed geodesic in a (closed) hyperbolic n-manifold is simple if it has no self-intersections, and nonsimple otherwise. In dimension 2 every closed hyperbolic manifold has a non-simple closed geodesic. However in dimension 3 the situation is much more complex. Many closed hyperbolic 3-manifolds contain immersions of totally geodesic surfaces and so there are non-simple closed geodesics. In [JR] examples were given of closed hyperbolic 3-manifolds containing a non-simple closed geodesic but having no immersed totally geodesic surface. However it was shown in [CR] that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert [BW] showed that almost all 3-manifolds arising as the quotient of H 1991 Mathematics Subject Classification. Primary: 57M50, Secondary: 30F40.

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).

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].