## Notes of Grigori Avramidi’s Cambridge lecture 23-06-2017

Topology of ends of nonpositively curved manifolds

Joint work with T. Nguyen Pham.

I am interested in complete Riemannian manifolds with curvature in ${[-1,0]}$, and finite volume.

Example. Product of two hyperbolic surfaces. The end is homeomorphic to ${N\times[0,+\infty)}$, with some extra structure: ${N}$ is made of two pieces.

More generally, for locally symmetric spaces of noncompact type, lifts of ends are homeomorphic to ${N\times[0,+\infty)}$, with ${N}$ a wedge of spheres. This description goes back to Borel-Serre.

1. Thick-thin decomposition

Gromov-Schroeder: assume there are no arbitrarily small geodesic loops. Then the thin part is homeomorphic to ${N\times[0,+\infty)}$, with ${N}$ a closed manifold.

The condition is necessary. Gromov gives an example of a nonpositively curved infinite type graph manifold of finite volume.

Theorem 1 (Avramidi-Nguyen Pham) Under the same assumptions, any map of a polyhedron to the thin part of the universal cover ${\tilde M}$ can be homotoped within the thin part into a map to an ${\lfloor \frac{n}{2}\rfloor}$-dimensional complex, ${n=dim(M)}$.

Consequences:

1. If ${n\leq 5}$, each component of the thin part is aspherical and has locally free fundamental group.
2. ${H^k(B\Gamma,{\mathbb Z} \Gamma)=0}$ for all ${k<\frac{n}{2}}$.
3. ${dim(B\Gamma)\geq \frac{n}{2}}$.

2. Proof

Maximizing the angle under which two visual boundary points are seen gives Tits distance, and the corresponding path metric ${Td}$.

In the universal cover, the thin part is the set of points moved less than ${\epsilon}$ away by some deck transformation ${\Gamma}$. Isometries are either hyperbolic (minimal displacement is achieved) or parabolic (infimal displacement is 0). Parabolic isometries have a nonempty fix-point set at infinity. At each point ${x}$, the subgroup generated by isometries moving ${x}$ no more than ${\epsilon}$ is virtually nilpotent, hence virtually has a common fixed point at infinity. This allows to define a discontinuous projection to infinity. The point is to show that the image has dimension ${<\lfloor \frac{n}{2}\rfloor}$.

2.1. Busemann simplices

If ${h_0}$ and ${h_1}$ are Busmeann functions, ${t_0h_0+t_1h_1}$ need not be a Busemann function again, but on each sphere, there is a unique point where it achieves its minimum, and tis point depends in a Lipschitz manner on ${t_0,t_1}$. This defines an arc in Tits boundary, hence simplices ${\sigma}$. We claim that

$\displaystyle \begin{array}{rcl} hom-dim(Stab(\sigma))+dim(image(\sigma))\leq n-1. \end{array}$

## Notes of Christopher Leininger’s Cambridge lecture 23-06-2017

Free-by-cyclic groups and trees

Joint work with S. Dowdall and I. Kapovich.

The Bieri-Neumann-Strebel invariant is an open subset ${\sigma G}$ of ${H^1(G)=Hom(G,{\mathbb R})}$, it is the set of ${u}$ such that ${\omega_u^{-1}({\mathbb R}_+)\rightarrow \hat X}$ is surjective on ${\pi_1}$. Here, ${\hat X}$ is the torsion free abelian cover of ${X=BG}$ and ${\omega_u}$ is an equivariant map ${\hat X\rightarrow{\mathbb R}}$ representing ${u}$.

If ${G}$ is free-by-cyclic, one can refine

$\displaystyle \begin{array}{rcl} \Sigma_{\mathbb Z} G=\{u\in\Sigma G\,;\,u(G)={\mathbb Z}\}. \end{array}$

Geoghegan-Mihalik-Sapir-Wise show that for every ${u\in \Sigma_{\mathbb Z} G}$, ${ker(u)}$ is locally free and there exists an outer automorphism ${\phi_u}$ and a finitely generated subgroup ${Q_u such that ${G=Q_u *_{\phi_u}}$. In particular, if ${u\in \Sigma_{\mathbb Z} G\cap(-\Sigma_{\mathbb Z} G)}$, then one can take ${Q_u=ker(u)}$.

From now on, we assume that ${\phi}$ is atoroidal and fully irreducible. Then ${G}$ is hyperbolic, and there exists an expanding irreducible train track representative (Bestvina-Handel). Let ${X=X_f}$ be the mapping torus. It carries the suspension of ${\phi}$, which is a one-sided flow (action of semi-group ${({\mathbb R}_+,+)}$). The representative ${\omega_u}$ of integral cohomology class ${u}$ factors to a map ${X\rightarrow S^1}$. Let ${S\subset H^1(G)}$ be the subset of cohomology classes ${u}$ such that the representative can be chosen to be increasing along the flow. Then

Theorem 1

1. ${S}$ is a component of ${\Sigma G}$. It is a rational polyhedral cone.
2. For ${u\in S_{\mathbb Z}}$, inverse images of points are cross-sections ${\Gamma_u}$ of the flow. The first return map ${f_u}$ is an expanding irreducible train track representative of ${\phi_u:Q_u\rightarrow Q_u}$, with ${\lambda(f_u)=\lambda(\phi_u)}$.

Stretch factors ${\lambda(f_u)}$ form a nice function on ${S}$.

Theorem 2 (Algom-Kfir-Hironaka-Rafi) There exists an ${{\mathbb R}}$-analytic, convex function ${h:S\rightarrow{\mathbb R}}$ such that for all ${u\in S}$ such that for al ${u\in S}$ and ${t>0}$,

1. ${\lim_{u\rightarrow\partial S}h(u)=+\infty}$.
2. ${h(tu)=\frac{1}{t}h(u)}$.
3. If ${u\in S_Z}$, then ${h(u)=\log(\lambda(f_u))=\log(\lambda(\phi_u))}$.

## Notes of Kevin Shreve’s Cambridge lecture 23-06-2017

Action dimension and ${L^2}$ Cohomology

Joint work with Giang Le and Mike Davis.

1. Action dimension

This is the minimal dimension ${actdim(G)}$ of contractible manifolds which admit a proper ${G}$-action. The geometric dimension ${gdim(G)}$ replaces manifolds with complexes.

1.1. Examples

If ${G}$ is of type ${F}$, then ${actdim(D)\leq 2 gdim(G)}$. This comes from embedding complexes ${BG}$ into ${{\mathbb R}^N}$. ${N=2n+1}$ would be easy. ${N=2n}$ is Stallings’ theorem, using a suitable model of ${BG}$.

Bestvina-Feighn: For lattices in semi-simle Lie groups, ${actdim(G)}$ is the dimension of the symmetric space.

Desputovic: ${actdim(MCG)=dim(}$Teichmuller space${)}$.

1.2. Our favourite examples

Today, we focus on graph products of fundamental groups of closed aspherical manifolds and complements of hyperplane arrangements. We are concerned with lower bounds: when can one reduce from the obvious dimension?

The first class (circles) includes RAAG, covered by Avramidi-Davis-Okun-Shreve.

1.3. Motivation from ${L^2}$-cohomology

Let ${b_i(\tilde M)}$ denote the ${L^2}$-Betti numbers of the universal covering.

Singer conjecture: If ${M}$ is a closed aspherical manifold of dimension ${n}$, then ${b_i(\tilde M)}$ vanish if ${i\not=n/2}$.

This suggests

Action dimension conjecture. If ${b_i(G)\not=0}$, then ${actdim(G)\geq 2i}$.

Okun and I have shown that both conjectures are in fact equivalent.

2. Graph products

Let ${L}$ be a flag complex with vertex set ${S}$. The graph product of a family ${\{G_s\,;\,s\in S\}}$ of groups over ${L}$ is the quotient of the free product of ${G_s}$ by the normal subgroup generated by ${[g_s,g_t]}$, when ${st}$ is an edge of ${L}$.

Examples. If all ${G_s={\mathbb Z}}$, we get RAAG. If all ${G_s}$ are finite cyclic, we get RACG.

Theorem 1 Let ${L}$ be a ${d-1}$-dimensional flag complex, let ${G_L}$ be the corresponding graph product of fundamental groups of closed aspherical ${m}$-manifolds. Then

1. If ${H_{d-1}(L,{\mathbb Z}_2)\not=0}$, then ${actdim(G_L)=md+d}$.
2. If ${H_{d-1}(L,{\mathbb Z}_2)=0}$, then ${actdim(G_L).

2.1. Constructing aspherical manifolds

The only way to make new aspherical manifolds is to glue aspherical manifolds with boundary along codimension 0 submanifolds of their boundaries. For instance, Salvetti complexes, made of tori, do not work. We replace tori with tori ${\times}$ interval.

In general, we glue together products of ${M_v\times I}$, which is ${md+d}$ dimensional, which is sharp in some cases, as we show next. The fact that ${L}$ has vanishing homology allows to decrease dimension.

2.2. Obstructions to actions

Bestvina-Kapovitch-Kleiner coarsify van Kampen’s obstruction to embedding complexes ${K}$ into ${{\mathbb R}^N}$. This lives in ${H^n(Conf_2(K),{\mathbb Z}_2)}$ (configuration of pairs of points).

Theorem 2 (Bestvina-Kapovitch-Kleiner) Let ${G}$ be ${CAt(0)}$ or hyperbolic, let ${K\subset\partial G}$ with ${vK^n(K)\not=0}$. Then

$\displaystyle \begin{array}{rcl} actdim(G)\geq n+2. \end{array}$

Example. If ${G=F_2\times F_2}$, ${\partial G}$ contains ${K_{3,3}}$, hence ${actdim(G)\geq 4}$ (in fact, ${=2}$).

For graph products of closed aspherical manifolds, we construct a complex, denoted by ${\hat O L}$, in ${\partial G_L}$. It is a join of ${m-1}$-spheres based on ${L}$.

## Notes of Bill Goldman’s Cambridge lecture 23-06-2017

The dynamics of classifying geometric structures

1. Marked geometric structures

Moduli spaces of geometric structures do not all behave like the moduli space of Riemann surfaces: in general, it is not a well behaved space, it is a quotient by a group action with interesting dynamics.

Lie and Klein (1872), Ehresmann (1936) suggest to study ${(G,X)}$-structures on manifolds ${S}$. Experience shows that it is useful to introduce a deformation space of marked ${(G,X)}$-structures, on which the mapping class group ${\pi_0(Diff(S))}$ acts. A marking is the data of a ${(G,X)}$-manifold ${S'}$ and a diffeomorphism ${S\rightarrow S'}$.

In some cases (e.g. hyperbolic structures on surfaces), this action is properly discontinuous, resulting in a quotient space which is a manifold mere singularities. In general, it is not.

1.1. Example: complete affine surfaces

All Euclidean structures on the 2-torus are affinelu isomorphic. Other affine structures, discovered by Kuiper, are obtained from the polynomial diffeomorphism

$\displaystyle \begin{array}{rcl} (x,y)\mapsto (x+y^2,y). \end{array}$

Indeed, change of charts turn out to be affine.

The mapping class group ${Gl(2,{\mathbb Z})}$ acts ergodically on the deformation space (Moore 1966).

2. Moduli spaces of representations

Let ${S}$ be a closed surface, ${\pi=\pi_1(S)}$. Let ${G}$ be a simple Lie group. Connected components of ${Rep(\pi,G)}$ are indexed by ${\tau\in\pi_1([G,G])}$.

With Forni, we try to use Teichmuller dynamics, and replace the difficult ${MCG}$ action by a simpler ${{\mathbb R}}$ action. This is defined on the unit tangent bundle of Teichmuller space ${T(S)}$.

Let ${E=(T(S)\times Rep(\pi,G)_\tau)/MCG}$. This is a bundle over . Let ${U}$ be its unit tangent bundle.

Theorem 1 (Forni-Goldman) For ${G}$ compact, the Teichmuller flow is strongly mixing on ${U}$.

Each element of ${\pi}$ defines a character function, hence a Hamiltonian flow. Dehn twists suffice to generate the ring of functions, hence

2.1. An example: compact surfaces of Euler characteristic ${-1}$

There are 4 of them, all have ${\pi_1=F_2}$. ${Rep(\pi,Sl(2))/Sl(2)}$ was determined as early as 1889. It is isomorphic (as a complex manifold) to ${{\mathbb C}^3}$.

The function ${k=Tr([\rho(X),\rho(Y)])}$ is invariant under ${Out(F_2)}$ (Nielsen). Level sets have invariant symplectic structures. Interesting involutions arise as deck transformations of branched double coverings given by coordinate projections to ${{\mathbb C}^2}$.

Level sets for values in ${(-2,2)}$ contain a component corresponding to unitary representations, on which the ${Out(F_2)}$ action is ergodic.

The case of the once-punctured Klein bottle is particularly interesting. The ${Out(F_2)}$ action does not extend to projective space.

## Notes of Denis Osin’s Cambridge lecture 22-06-2017

Extending group actions on metric spaces

Joint work with David Hume and C. Abbott.

Question. Let ${H be groups. Given an isometric action of ${H}$ on a metric space ${S}$, does it extend to an action on a (possibly different) metric space ?

1. Extensions of actions

What to be mean by extension? We have in mind induction of representations.

Let ${H}$ act on ${S}$ and ${R}$. Say that a map ${f:S\rightarrow R}$ is coarsely equivariant if for every ${x\in R}$, ${h\mapsto d(f(hx),hf(x))}$ is bounded on ${H}$.

Definition 1 Say an action of group ${G}$ on ${R}$ is an extension of the action of subgroup ${H}$ on ${S}$ is there exists a coarsely ${H}$-equivariant quasi-isometric embedding ${f:S\rightarrow R}$.

Definition 2 We say that the extension problem (EP) for ${H is solvable if every action of ${H}$ on a metric space extends to an action of ${G}$.

1.1. Examples

This is rather flexible.

1. If ${H}$ has bounded orbits, the trivial action of ${G}$ is an extension.
2. If ${H}$ is a retract of ${G}$ (i.e. there exists a homomorphism ${G\rightarrow H}$ which is the identity on ${H}$), then every actions of ${H}$ extends.
3. Fix finite generating systems of ${H}$ and ${G}$. Assume ${H}$ is undistorted in ${G}$. Then the action of ${H}$ on its Cayley graph extends to the action of ${G}$ on its Cayley graph.
4. An example where (EP) is not solvable. Let ${G=Sym({\mathbb N})}$. Then every action of ${G}$ on a metric space has bounded orbits (Cornulier). If ${H, no action of ${H}$ with unbounded orbits can extend.
5. A converse of (3) holds: if ${G}$ is finitely generated and (EP) is solvable for ${H then ${H}$ is finitely generated and undistorted in ${G}$. Whence many examples where (EP) is not solvable. Furthermore, if ${G}$ is finitely generated and elementarily amenable, then (EP) is solvable for all ${H implies that ${G}$ is virtually abelian.
6. Let ${H=F_2}$ be a free group and ${G=H *_\phi}$ where ${\phi}$ exchanges generators. Then translation action of ${H}$ on ${{\mathbb R}}$ with one generator acting trivially cannot extend to ${G}$. Indeed, one generator of ${H}$ has bounded orbits, the other does not, but both are conjugate in ${G}$.

1.2. Hyperbolic embeddings

The following definition appears in Dahmani-Guirardel-Osin. Let ${X\subset G}$ be a subset such that ${X\cup H}$ generates ${G}$. Let ${\hat d}$ be the metric on ${H}$ induced by the embedding of ${H}$ (as vertex set of complete graph ${Cay(H,H)}$) into ${Cay(G,X\cup H)}$ with edges of ${Cay(H,H)}$ removed. Say that ${H}$ is hyperbolically embedded in ${(G,X)}$ if

1. ${Cay(G,X\cup H)}$ is hyperbolic,
2. ${(H,\hat d)}$ is proper.

For instance,

1. ${H}$ is not hyperbolically embedded into ${H\times{\mathbb Z}}$ , but it is into ${H*{\mathbb Z}}$.
2. Observe that there exists a finite subset ${X\subset G}$ such that ${H}$ is hyperbolically embedded into ${(G,X)}$ iff ${G}$ is hyperbolic relative to ${H}$.
3. If ${a\in MCG}$ is pseudo-Anosov, then there exists a virtually cyclic subgroup ${E}$ containing ${a}$ which is hyperbolically embedded in ${MCG}$.

1.3. Acylindrically hyperbolic groups

This class contains ${MCG}$, ${Out(F_n)}$, finitely presented groups of deficiency ${\geq 2}$ (argument uses ${\ell^2}$-Betti numbers).

Theorem 3 (Dahmani-Guirardel-Osin) If ${G}$ is acylindrically hyperbolic, then it contains hyperbolically embedded subgroups of the form ${F_n\times}$ finite for all ${n}$.

2. Results

Theorem 4 Let ${H be hyperbolically embedded. Then (EP) is solvable for ${H. Moreover, every action of ${H}$ on a hyperbolic metric space extends to a action of ${G}$ on a hyperbolic metric space.

Corollary 5 Let ${G}$ be a hyperbolic group, and ${H.

1. If ${H}$ is virtually cyclic, then (EP) is solvable for ${H.
2. If ${H}$ is quasi-convex and almost malnormal (${|H\cap H^g|<\infty}$ for all ${g\notin H}$), then (EP) is solvable for ${H.
3. Conversely, if (EP) for ${H is solvable, then ${H}$ is quasi-convex.

## Notes of Jean-François Lafont’s Cambridge lecture 22-06-2017

Hyperbolic groups whose boundary is a Sierpinski ${n}$-space

Joint work with Bena Tshishiku.

1. Sierpinski ${n}$-space

Start with an ${(n+1)}$-dimensional sphere. Remove a dense family of balls with disjoint interiors. Get ${S_n}$. Up to homeo, balls need not be round. One merely needs that their diameters tend to 0.

Any homeo of ${S_n}$ permutes the distinguished peripheral spheres.

Examples.

1. Free groups have ideal boundary ${S_0}$.
2. If ${M}$ is a compact negatively curved ${n}$-manifold with nonempty totally geodesic boundary, then ${\partial\tilde M=S_{n-2}}$.
3. Let ${\Gamma}$ be a nonuniform lattice of isometries of ${H^n}$. Then ${\Gamma}$ is cocompact on the complement ${A}$ of a union of horospheres, hence ${\partial A=S_{n-2}}$.

1.1. Cannon conjecture

What properties of the group follow from specifying the topology of the boundary ? This is what Cannon’s conjecture is about: if ${\partial\Gamma=S^2}$, must ${\Gamma}$ be a cocompact lattice in ${H^3}$?

Here is a topological variant of Cannonc’s conjecture.

Theorem 1 (Bartels-Lueck-Weinberger) If ${\Gamma}$ is torsion-free hyperbolic, and ${\partial \Gamma=S^{n-1}}$, and ${n\geq 6}$, then there exists a unique closed aspherical ${n}$-manifold ${M}$ with ${\Gamma=\pi_1(M)}$.

2. Result

Theorem 2 If ${\Gamma}$ is torsion-free hyperbolic, and ${\partial \Gamma=S_{n-2}}$, and ${n\geq 7}$, then there exists a unique aspherical ${n}$-manifold ${M}$ with nonempty boundary with ${\Gamma=\pi_1(M)}$. Moreover, every boundary component of ${M}$ corresponds to a quasi-convex subgroup of ${\Gamma}$.

3. Proof

3.1. Step 1

Kapovitch-Kleiner: ${\Gamma}$ is a relative ${PD(n)}$-group, relative to the collection of stablizers of peripheral spheres.

3.2. Step 2

Realize ${\Gamma}$ as ${\pi_1(X)}$, where ${X}$ is a finite relative ${PD}$ complex, relative to a finite subcomplex ${Y\subset X}$. We use the Rips complex for ${B\Gamma}$ but the Bartles-Lueck-Weinberger complexes for parabolic subgroups ${\Lambda_i}$.

3.3. Surgery theory

Browder-Novikov-Sullivan-Wall surgery theory provides obstructions to finding a manifold homotopy equivalent to ${X}$. They belong to the space ${S(X)}$ that appears in the algebraic surgery exact sequence

$\displaystyle \begin{array}{rcl} \cdots\rightarrow H_n(X,L_\cdot)\rightarrow H_n({\mathbb Z}\Gamma)\rightarrow S(X)\rightarrow H_{n-1}(X,L_\cdot)\rightarrow\cdots. \end{array}$

A similar exact sequence appears in 4-periodic surgery exact sequence, with ${L}$ replaced with a very similar ${\bar L}$ (and ${S(X)}$ with ${\bar S(X)}$). They have the same homotopy groups and differ only in their 0-spaces

$\displaystyle \begin{array}{rcl} \textrm{for }L_\cdot, ~G/TOP ; \quad \textrm{for }\bar L_\cdot, ~G/TOP\times L_0({\mathbb Z}). \end{array}$

There is a long exact sequence

$\displaystyle \begin{array}{rcl} \cdots\rightarrow H_n(X,L_0({\mathbb Z}))\rightarrow S_n(X)\rightarrow\bar S_n(X)\rightarrow H_{n-1}(X,L_0({\mathbb Z}))\rightarrow\cdots. \end{array}$

It turns out that ${H_n(X,L_0({\mathbb Z}))=H_n(X,{\mathbb Z})=0}$. Furthermore, thanks to the (L-theoretic) Farrell-Jones isomorphism conjecture (which holds for hyperbolic groups, Bartels-Lueck-Reich), ${\bar S_n(X)=0}$. Hence ${S_n(X)=0}$, the obstruction vanishes, so there exists a homology manifold model for ${B\Gamma}$.

3.4. ${CAT(0)}$ groups

Bartels-Lueck-Reich cover ${CAT(0)}$ groups. In the relative case (replace spheres with Sierpinski spaces), much of the argument carries over, but the first step.

## Notes of Michelle Bucher’s Cambridge lecture 22-06-2017

Vanishing simplicial volume for certain affine manifolds

Joint work with C. Connell and J.-F. Lafont.

1. Affine manifolds

An affine manifold is a compact smooth ${n}$-manifold equipped with an atlas of maps to ${{\mathbb R}^n}$ with coordinate changes in the affine group ${{\mathbb R}^n\times Gl(n,{\mathbb R})}$. An affine manifold has a developing map ${\tilde M\rightarrow{\mathbb R}^n}$. An affine manifold is complete if every affine segment can be extended forever. Equivalently, if the developing map is 1-1 (the ${M}$ is a quotient of ${{\mathbb R}^n}$ by a discrete group of affine transformations).

1.1. Examples

Quotients, like the circle ${S^1={\mathbb R}/{\mathbb Z}}$.

Mapping tori of affine toral automorphisms.

Products of hyperbolic manifolds with the circle (modelled on cone ${H^n\times{\mathbb R}}$, where ${H^n}$ is the pseudosphere in ${{\mathbb R}^{n,1}}$.

1.2. Famous problems

Auslander conjecture. The fundamental group of a complete affine manifold is virtually solvable.

Known in low dimensions (Fried-Goldman if ${n=3}$, Abels-Margulis-Soifer for ${n=4,5,6}$).

Chern conjecture. The Euler characteristic of every affine manifold vanishes.

Known in 2 dimensions (Benzecri), in the complete case (Koslark-Sullivan), when there is an invariant volume (Klingler), for irreducible higher rank locally symmetric spaces (Margulis), local products of surfaces (Bucher-Gelander), connected sums of manifolds with finite fundamental groups (Smillie).

Markus conjecture. An affine manifold is complete iff it has an invariant volume.

2. Simplicial volume

This is the ${\ell^1}$ norm of the fundamental class (Gromov).

For every continuous map ${f:M\rightarrow N}$ between ${n}$-manifolds,

$\displaystyle \begin{array}{rcl} |deg(f)|\|M\|\leq \|N\|, \end{array}$

hence ${\|M\|=0}$ for tori, for instance. On the other hand, for a higher genus surface, ${\|M\|=-2\chi(M)}$. More generally, for a hyperbolic ${n}$-manifold, ${vol(M)=v_n\|M\|}$, where ${v_n}$ is the volume of the regular ideal simplex in hyperbolic ${n}$-space (Gromov-Thurston). Similar formulae are conjectured for other locally symmetric spaces, but the only known cases is ${\|M\|=6\chi(M)}$ for quotients of ${H^2\times H^2}$ (Bucher).

Theorem 1 (Sullivan-Smillie) For affine ${n}$-manifolds, ${|\chi(M)|\leq 2^{-n}\|M\|}$.

In fact, all what is used is the flatness of the tangent bundle. The proof constructs a PL vectorfield with at most one zero per simplex of a triangulation.

This suggests a strengthening of Chern’s conjecture: does ${\|M\|=0}$ for all affine manifolds?

2.1. Result

Theorem 2 Let ${M}$ be an aspherical affine manifold. Assume that

1. the holonomy representation is faithful,
2. its image contains a pure translation.

Then ${\|M\|=0}$.

2.2. Proof

Lueck asked wether every aspherical manifold whose fundamental group has a nontrivial normal amenable subgroup ${A}$ has vanishing simplicial volume. We give a partial answer.

Lemma 3 This is true if the map

$\displaystyle \begin{array}{rcl} H^n(\pi_1(M)/A)\rightarrow H^n(\pi_1(M)) \end{array}$

vanishes.

The point is to show that the map ${H_b^n(M)\rightarrow H^n(M)}$ is not onto. Amenability implies that ${H^n(\pi_1(M)/A)\rightarrow H_b^n(\pi_1(M))}$ is an isomorphism. A commutative diagram show that ${H_b^n(M)\rightarrow H^n(M)}$ vanishes.

We apply the Lemma to the kernel of linear holonomy ${A=ker(\pi_1(M)\rightarrow Gl(n,{\mathbb R}))}$, which by assumption, maps injectively to translations of ${{\mathbb R}^n}$.