Amenable Fundamental Group Research Articles

Simplicial volume of manifolds with amenable fundamental group at infinity

Abstract We show that for $n \neq 1,4$ , the simplicial volume of an inward tame triangulable open $n$ -manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

Variations on the theme of the uniform boundary condition

The uniform boundary condition (UBC) in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the [Formula: see text]-norm on the singular chain complex, Matsumoto and Morita established a characterization of the UBC in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the UBC in every degree. We will give an alternative proof of statements of this type, using geometric Følner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.

2-dimensional polyhedra with finite depth

In this paper every polyhedron is finite and every ANR is compact. Let P≥X1≥X2≥…, be a sequence, in which P is a polyhedron and ≥ are homotopy dominations. One may ask, if each sequence of this form contains only finitely many homotopy dominations that are not homotopy equivalences, or if there exists an integer lP (independent of the sequence) that each sequence contains only ≤lP homotopy dominations that are not homotopy equivalences. Closely related open questions are: Does there exist a polyhedron P homotopy dominating an infinite sequence of polyhedra{Pi}, wherePihomotopy dominatesPi+1butPiandPi+1have different homotopy types, for everyi∈N? (M. Moron) [30, Problem 1436], and the famous problem of K. Borsuk (1967): Is it true that twoANR′shomotopy dominating each other have the same homotopy type? Here we prove that, if dim⁡P=2, then the answers to all these questions depend only on the properties of the fundamental group of P (for 1-dimensional polyhedra, the answers are obvious). Furthermore, if sequences in consideration contain only polyhedra, then, for the positive answer it suffices to answer positively the analog of our topological question for finitely presented groups (the fundamental groups) with retractions. Applying these results, we prove that for each polyhedron P with dim⁡P≤2 and elementary amenable fundamental group G with cdG<∞, there is a bound lP on the lengths of all descending sequences P≥X1≥X2… of homotopy dominations that are not homotopy equivalences. The same holds if the fundamental group of P is a limit group. It means that such polyhedra P have finite depth.

Amenable groups and smooth topology of 4-manifolds

A smooth five-dimensional [Formula: see text]-cobordism becomes a smooth product if stabilized by a finite number [Formula: see text] of [Formula: see text]’s. We show that for amenable fundamental groups, the minimal [Formula: see text] is subextensive in covers, i.e. [Formula: see text]. We focus on the notion of sweepout width, which is a bridge between [Formula: see text]-dimensional topology and coarse geometry.

On the equivalence of the definitions of volume of representations

Let G be a rank 1 simple Lie group and M be a connected orientable aspherical tame manifold. Assume that each end of M has amenable fundamental group. There are several definitions of volume of representations of the fundamental group of M into G. We give a new definition of volume of representations and furthermore, show that all definitions so far are equivalent.

Simplicial volume of compact manifolds with amenable boundary

Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on M whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

Isometric embeddings in bounded cohomology

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups, we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X, Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X, Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov's Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along π1-injective boundary components with amenable fundamental group.

Macroscopic dimension and essential manifolds

M. Gromov asked whether the macroscopic dimension of rationally essential n-dimensional manifolds equals n. We show that the answer depends only on the corresponding group homology class and give an affirmative answer for certain classes. In particular, the answer is positive for manifolds with amenable fundamental groups.

The limit of $\mathbb{F}_{p}$-Betti numbers of a tower of finite covers with amenable fundamental groups

We prove an analogue of the Approximation Theorem of -Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.

Closed orbits of a charge in a weakly exact magnetic field

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $\sigma$ a weakly exact 2-form. Let $\phi_{t}$ denote the magnetic flow determined by $\sigma$, and let $c$ denote the Mane critical value of the pair $(g,\sigma)$. We prove that if $k>c$, then for every non-trivial free homotopy class of loops on $M$ there exists a closed orbit with energy $k$ whose projection to $M$ belongs to that free homotopy class. We also prove that for almost all $k<c$ there exists a closed orbit with energy $k$ whose projection to $M$ is contractible. In particular, when $c=\infty$ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that if $\sigma$ is not exact and $M$ has an amenable fundamental group (which implies $c=\infty$) then there exist contractible closed orbits on almost every energy level.

Classification of negatively pinched manifolds with amenable fundamental groups

We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the Möbius band, and the products of R with the total spaces of flat vector bundles over closed infranilmanifolds.

On Complex Codimension-One Foliations Transverse Fibrations

We study codimension-one holomorphic foliations which are transverse to fibrations and, therefore, are conjugate to suspensions of groups of Mobius transformations. For the general case, we assume that there exists an invariant transverse measure. The existence of such an invariant transverse measure is proved for the case where there is a leaf with the subexponential growth or the basis has an amenable fundamental group. This result applies, e.g., for the study of amenable Lie group actions on complex projective spaces.

Liouville property for groups and manifolds

We introduce a method to estimate the entropy of random walks on groups. We apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not Liouville. We also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non-trivial Poisson boundary if and only if this group is not virtually nilpotent. This, in particular, shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient Lie group. As another application we get a series of examples of amenable groups such that any finite entropy non-degenerate measure on them has non-trivial Poisson boundary. Since the groups in question are amenable, they do admit measures such that the corresponding random walks have trivial boundary; the above shows that such measures on these groups have infinite entropy.

The Chern Conjecture for Affinely Flat Manifolds Using Combinatorial Methods

An affine manifold is a manifold with a flat affine structure, i.e. a torsion-free flat affine connection. We slightly generalize the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is zero confirming an old conjecture of Chern. The technique is from Kim and Lee's work using the combinatorial Gauss–Bonnet theorem and taking the means of the angles by amenability. We show that if an even-dimensional manifold is obtained from a connected sum operation from K(π, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure generalizing a result of Smillie.

Embedding Seifert Fibred 3-Manifolds and Sol3 -Manifolds in 4-Space

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds which may embed as topologically locally flat submanifolds of S4. In particular, a circle bundle over a non-orientable surface F embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of F in S4, and we show that precisely thirteen geometric 3-manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0-framed surgery on a link which is the union of two slice links, and so embeds smoothly in S4. 1991 Mathematics Subject Classification: 57N13.

Manifolds of even dimension with amenable fundamental group

Manifolds of even dimension with amenable fundamental group

The Euler Characteristic of Projectively Flat Manifolds with Amenable Fundamental Groups

The Euler characteristic of a closed projectively flat manifold with amenable fundamental group is shown to be nonnegative, and in fact zero if we further assume that the developing map is injective and the fundamental group is infinite.

The Euler characteristic of projectively flat manifolds with amenable fundamental groups

The Euler characteristic of a closed projectively flat manifold with amenable fundamental group is shown to be nonnegative, and in fact zero if we further assume that the developing map is injective and the fundamental group is infinite.