A sharper tits alternative for 3-manifold groups

The following theorem is proven. LetM be a closed, orientable, irreducible 3-manifold such that rankH ₁(M, ℤ/pℤ)≥3 for some primep. Then either π₁(M) is virtually solvable or it contains a free group of rank 2.     

Almost convex groups and the eight geometries

IfM is a closed Nil geometry 3-manifold then ₁(M) is almost convex with respect to a fairly simple geometric generating set. IfG is a central extension or a extension of a word hyperbolic group, thenG is also almost convex with respect to some generating set. Combining these with previously known results shows that ifM is a closed 3-manifold with one of Thurston's eight geometries, ₁(M) is almost convex with respect to some generating set if and only if the geometry in question is not Sol.     

Computing the genus of the 2-amalgamations of graphs

The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG ₁ andG ₂ (namely ifG=G ₁∪G ₂ andG ₁∩G ₂={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G ₁)+γ(G ₂)+ε, whereε=0,1 or −1. In this paper we sharpen that result by giving a means by whichε may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.     

Combings of semidirect products and 3-manifold groups

IfG is a finitely generated group that is abelian or word-hyperbolic andH is an asynchronously combable group then every split extension ofG byH is asynchronously combable. The fundamental group of any compact 3-manifold that satisfies the geometrization conjecture is asynchronously combable. Every split extension of a word-hyperbolic group by an asynchronously automatic group is asynchronously automatic.     

Profinite properties of graph manifolds

Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π ₁(M) is efficient with respect to the JSJ decomposition of M. We go on to prove that π ₁(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π ₁(M) is conjugacy separable.     

Margulis numbers for Haken manifolds

For every closed orientable hyperbolic Haken 3-manifold and, more generally, for any orientable hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If H ₁(M;ℚ) ≠ 0, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

Tits alternative for 3-manifold groups

Let M be an irreducible, orientable, closed 3-manifold with fundamental group G. We show that if the pro-p completion of G is infinite then G is either soluble-by-finite or contains a free subgroup of rank 2. Both authors are partially supported by “Bolsa de produtividade de pesquisa” from CNPq, Brazil. Received: 16 February 2006     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)&\ne;0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999     

Laminated decompositions involving a given submanifold

Let M denote a connected (n+1)-manifold (without boundary). We study laminated decompositions of M, by which we mean upper semicontinous decompositions G of M into closed, connected n-manifolds. In particular, given M with a lamination G and N, a locally flat, closed, n-dimensional submanifold, we determine conditions under which M admits another lamination G_N with N?G_N. For n ≠ 3 a sufficient condition is that i: N → M be a homotopy equivalence. For n > 3 we give examples to show that i: N → M being a homology equivalence is not sufficient. We also show how to replace the assumption of local flatness of N with a weaker cellularity criterion (n ? 4) known as the inessential loops condition. We then give examples illustrating the abundance of pathology if M is not assumed to have a preexisting lamination.     

Nielsen’s theorem for model aspherical manifolds

The main issue of this paper is the discussion of Nielsen’s realisation-problem for aspherical manifolds arising from (generalised) Seifert fiber space constructions. We present sufficient conditions on such “model” aspherical manifoldsM to have that a finite abstract kernel ψ:G → Out (π₁ (M)) can be (effectively) geometrically realised by a group of fiber preserving homeomorphisms ofM if and only if ψ can be realised by an (admissible) group extension 1 → (π₁ (M)) →E’ →G → 1. Then an algebraic approach to a (partial) study of the symmetry ofM is possible. Our result covers all situations already described in literature and we show with an example that we also deal with other types of Seifert fiber space constructions which were not yet treated before. Research Assistant of the Belgian National Fund for Scientific Research (N.F.W.O.)     

Peripheral fillings of relatively hyperbolic groups

In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π₁(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26     

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

A primary obstruction to topological embeddings¶and its applications

For a proper continuous map f:M→N between topological manifolds M and N with m≡ dimM < dimN≡m+k, a primary obstruction to topological embeddings θ(f) ∈H ^c _{m − k} (M; Z ₂) has been defined and studied by the authors in {9, 8, 2, 3], where H ^c _* denotes the singular homology with closed support. In this paper, we study the obstruction from the viewpoint of differential topology and give various applications. We first give some characterizations of embeddings among generic differentiable maps, which are refinements of the results in [9, 10]. Then we give a result concerning the number of connected components of the complement of the image of a codimension-1 continuous map with a normal crossing point, which generalizes the results in [6, 4, 5, 9]. Finally we give a simple proof of a theorem of Li and Peterson [20] about immersions of m-manifolds into (2m-1)-manifolds. Received: 3 December 1999 / Revised version: 10 October 2000     

Lefschetz fixed point theory on fibrations

For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H ⁰(E) and H ⁰(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π ₁ (G/H) = 0.