Notes of Gang Tian’s Orsay lecture 28-06-2017

Existence of conic Kaehler-Einstein metrics

Joint work with Feng Wang, Zhejiang university.

A log-Fano manifold is the date of a compact Kaehler manifold {M}, a divisor with normal crossings {D=\sum(1-\beta_i)D_i} such that the line bundle

\displaystyle  \begin{array}{rcl}  L=K_M^{-1}+\sum(1-\beta_i)D_i \end{array}

is positive.

A metric {\omega} is a conic Kaehler-Einstein metric if it is smooth Kaehler in {M\setminus |D|} and for every point {p\in|D|} where {D} is defined by {z_1\cdots z_d=0} in some coordinates, {\omega} is equivalent (between two multiplicative constants) to the model cone metric

\displaystyle  \begin{array}{rcl}  \omega_{cone}=i(\sum_{i\leq d} \frac{dz_i\wedge d\bar z_i}{|z_i|^{2(1-\beta_i)}}+\sum_{i>d} dz_i\wedge d\bar z_i). \end{array}

Say that {\omega} is conic Kaehler-Einstein if

\displaystyle  \begin{array}{rcl}  Ric(\omega)=\omega+2\pi\sum(1-\beta_i)[D_i]. \end{array}

1. Necessary conditions

Berman 2016: If {(M,D)} admits a conic KE metric with {[\omega]=2\pi c_1(L)}, then {(M,D)} is log-K-stable.

Log-K-stability is defined as follows.

A special degeneration {(\mathcal{X},\mathcal{D},\mathcal{L})} of {M,D,L)} is a 1-parameter family of log-pairs, consisting of

  1. A normal log-pair {(\mathcal{X},\mathcal{D})} with a {{\mathbb C}^*}-equivariant map {\pi:(\mathcal{X},\mathcal{D})\rightarrow{\mathbb C}},
  2. {\mathcal{L}} is an equivariant {\pi}-ample {{\mathbb Q}}-line bundle.
  3. {\mathcal{X}_t,\mathcal{D}_t,\mathcal{L}_t} is isomorphic to {(M,D,L)} for every {t\not=0}.

There is a natural compactification {(\overline{\mathcal{X}},\overline{\mathcal{D}},\overline{\mathcal{L}})} of {(\mathcal{X},\mathcal{D},\mathcal{L})} that maps to {{\mathbb C} P^1}. Defined number

\displaystyle  \begin{array}{rcl}  w(\mathcal{X},\mathcal{D},\mathcal{L})=\frac{n\overline{\mathcal{L}}^{n+1}+(n+1)\overline{\mathcal{L}}^n(K_{\mathcal{X}|{\mathbb C} P^1}+\overline{\mathcal{D}})}{(n+1)L^n}. \end{array}

If the central fiber is a log-Fano variety {(M_0,D_0)} embedded in {{\mathbb C} P^N} by {H^0(M_0,L^\ell_{|M_0})}, then {w(\mathcal{X},\mathcal{D},\mathcal{L})} can be interpreted as a Futaki invariant.

Say that {(M,D)} is log-K-semistable if for any special degeneration {(\mathcal{X},\mathcal{D},\mathcal{L})} has {w(\mathcal{X},\mathcal{D},\mathcal{L})\geq 0}. Say that {(M,D)} is log-K-stable if for any special degeneration {(\mathcal{X},\mathcal{D},\mathcal{L})} has {w(\mathcal{X},\mathcal{D},\mathcal{L})\geq 0} and equality holds only for the trivial degeneration {(\mathcal{\mathcal{X},\mathcal{D},\mathcal{L}})=(M,D,L)\times{\mathbb C}}.

2. The result

Theorem 1 If {(M,D)} is log-K-stable, the there exists a conic KE metric {\omega} with {[\omega]=2\pi c_1(L)}.

Many special cases were known, as consequences of existence of KE metrics on smooth closed manifolds. For instance when {D} is a multiple of {K_M^{-1}}.

3. Motivation

We are interested in {{\mathbb Q}}-Fano varieties {X}. Assume {X} admits a resolution {\mu:M\rightarrow X} such that {K_M=\mu^* K_X+\sum a_i E_i}, {a_i\in(-1,0]}. For small enough {\epsilon\in{\mathbb Q}}, define

\displaystyle  \begin{array}{rcl}  L_\epsilon=\pi^*K_X-\sum \epsilon E_i. \end{array}

If there exists a KE metric {\omega} on {X}, then {\mu^*\omega} is a degenerate conic KE metric on {M} with conic angles {2\pi b_i} along {E_i}. We expect that there exist conic KE metrics {\omega_\epsilon} on {(M,\sum (1-b_i-\epsilon)E_i)} with {[\omega_\epsilon]=2\pi c_1(L_\epsilon)}, which Gromov-Hausdorff converge to {(X,\omega)} as {\epsilon\rightarrow 0}.

We think that we are now able to prove the following. If {X} is a K-stable {{\mathbb Q}}-Fano variety. Then it admits a generalized KE metric in the above sense.

4. Proof

Many steps are similar to the smooth case. Pick a large integer {\lambda} such that {\lambda L} has a smooth divisor {E}. We use a continuity method, solving

\displaystyle  \begin{array}{rcl}  Ric(\omega_t)=t\omega_t+\frac{1-t}{\lambda}[E]+\sum(1-\beta_i)[D_i], \end{array}

{t\in[0,1]}. The set {I} of {t} such that a solution exists is easily shown to be non-empty (it contains 0) and open. Is it closed? The key point is a {C^0} estimate. It follows from a “partial {C^0}-estimate” and log-K-stability. In turn, this follows from an {L^2}-estimate and compactness a la Cheeger-Colding-Tian.

4.1. Smoothing conical KE metrics

Say that {\omega} has a K-approximation if there exist Kaehler metrics {\omega_i=\omega+i\partial\bar\partial\phi_i} in the same cohomology class such that

  • {\phi_i\rightarrow 0} uniformly on {M} and smoothly outside {|D|},
  • {Ric(\omega_i)\geq K\omega_i},
  • {(M,\omega_i)\rightarrow (M,\omega)} in Gromov-Hausdorff topology.

We show that if {Aut^0(M,D)=\{1\}} and if for all {i},

\displaystyle  \begin{array}{rcl}  (1-K_i)L+(1-\beta_i)D_i\geq 0 \end{array}

for some {K_i\leq 1}, then {\omega} has a K-approximation where {K=\sum(K_i-1)+1}.

We solve a modified equation with an extra term involving {K_i}‘s. For this, we use the variational approach by Boucksom-Eyssidieux-Guedj-Zeriahi and results of Darwan-Robinstein, Guenancia-Paun.

4.2. Extend B. Wang-Tian’s results to conic case

5. Work in progress

To handle {{\mathbb Q}}-Fano varieties, we need to extend Cheeger-Colding to conic cases.

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Notes of Oana Ivanovici’s Orsay lecture 28-06-2017

Geometry and analysis of waves in manifolds with boundary

The wave-front is a subset of the cotangent bundle, whose projection is the singular support. In all dimensions, in Euclidean space, it travels at constant speed along straight lines (Fermat,…, Hormander).

In general Riemannian manifolds without boundary, it travels along geodesics as long as time stays less than the injectivity radius (Duistermaat-Hormander).

We impose Dirichlet boundary conditions. Then transverse waves reflect according to Snell’s law of reflection (Chazarain). What about tangencies? Assume obstacle is convex. Do waves propagate in the shadow?

Melrose-Taylor 1975: if the boundary is {C^\infty}, no smooth singularities in the shadow region. However, analytic singularities occur.

Inside strictly convex domains, waves reflect a large number of times. The wave shrinks in size between two reflections, it refocusses, therefore its maximum increases. Caustics appear, together with swallowtail and cusp singularities.

In the non-convex case, especially if infinite order tangencies occur, one does not even know what the continuation of a ray should be (Taylor 1976).

1. Dispersive estimates

It is a measurement of the decay of amplitude of waves due to spreading out while energy is conserved.

In {{\mathbb R}^d}, after a high frequency cut-off around frequency {\lambda}, the maximum amplitude decays like {\lambda^{(d+1)/2}t^{-(d-1)/2}}. Indeed, the wave is concentrated in an annulus of width {1/\lambda}. The same holds in Riemannian manifolds without boundary.

In the presence of boundary, propagation of singularities has brought results in the 1980’s. Later on, people have tried a reduction to the boundary-less case with a Lipschitz metric: this requires no assumptions on the boundary, but ignores reflection and its refocussing effect.

1.1. Within convex domains

Theorem 1 (Ivanovici-Lascar-Lebeau-Planchon 2017) For strictly convex domains, dispersion is in

\displaystyle  \begin{array}{rcl}  \lambda^d\max\{1,(\lambda t)^{-\frac{d-1}{2}+\frac{1}{4}}\}. \end{array}

This follows from a detailed description of the wave-front, including swallow-tails. It takes into account infinitely many reflections. It is sharp.

1.2. Outside convex obstacles

The Poisson spot. This is a place where diffracted light waves interfere. It is in the shadow area, but much more light concentrates there. This was confirmed experimentally by Arago, following a debate launched by Fresnel who did not believe in the wave description of light. It should exist if one believes in Fermat’s principle that light rays follow geodesics, including those which creep along the boundary surface (Keller’s conjecture). In 1994, HargĂ© and Lebeau proved that, when light creeps along the bounday, it decays like {e^{-\lambda^{1/3}}}.

Theorem 2 (Ivanovici-Lebeau 2017) For strictly convex obstacles,

  1. if {d=3}, dispersion estimates hold like in {{\mathbb R}^3},
  2. if {d\geq 4}, they fail at the Poisson spot.

The reason is that a {d-2}-dimensional surface lits the Poisson point.

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Notes of Erlend Grong’s Orsay lecture 27-06-2017

Asymptotic expansions of holonomy

Joint with Pierre Pansu.

1. Motivation

Given a connection on a principal bundle {G\rightarrow P\rightarrow M}, holonomy along a based loop {\gamma} of {M} is an element of {G} resulting from lifting horizontally {\gamma} to {P}. We look for an expression {F(\gamma)\in\mathfrak{g}} such that {\exp(F(\gamma))} is a good approximation of holonomy when {\gamma} is short,

\displaystyle  \begin{array}{rcl}  hol(\gamma)=\exp(F+O(\ell^r)). \end{array}

We want that {F(\gamma)} be simpler to compute than holonomy, and be related to curvature.

Hatton-Choset: motion of a snake with two joins. {M=S^1\times S^1}, {G=SE(2)}. Experimentalists have been led to choose the Coulomb gauge, and for {F(\gamma)} the integral over a disk spanning {\gamma} of curvature expressed in Coulomb gauge.

In this practical example, motions are tangent to a sub-bundle of the tangent bundle of {M}. Hence our interest in expansions which are particularly efficient on such curves. We call this setting sub-Riemannian.

Sub-Riemannian curvature is not easy to define. The obvious approach of using adapted connections on the tangent bundle is not illuminating.

2. Results

  1. Asymptotic, gauge-free formula in Euclidean space.
  2. Riemannian case not that different.
  3. Sub-Riemannian case suggests a notion of curvature.
  4. For certain sub-Riemannian structures,

2.1. Euclidean case

Dilations define radial fillings {disk(\gamma)} of loops. Use radial gauge (frame is parallel along rays through the origin). They turn out to be optimal. Using radial gauge, integrate curvature over radial filling. This defines

\displaystyle F(\gamma)=\int_{\mathrm{disk}(\gamma)}\Omega.

Say a differential form {\omega} has weight {\geq m} if dilates {\delta_s^*\omega} are {O(t^m)}. Use radial gauge to define weight of forms on {P}.

Theorem 1 If the curvature has weight {m}, then

\displaystyle  \begin{array}{rcl}  hol(\gamma)=\exp(F(\gamma)+O(\ell(\gamma)^2m). \end{array}

Furthermore, one can expand {F(\gamma)} in termes of Taylor’s expansion of curvature.

Since curvature has weight at least 2, one gets a 4-th order approximation.

2.2. Sub-Riemannian case

The flat sub-Riemannian case corresponds to Carnot groups, i.e. a Lie group whose Lie algebra has a gradation

\displaystyle  \begin{array}{rcl}  \mathfrak{n}=\mathfrak{n}_1\oplus\cdots\oplus \mathfrak{n}_r \end{array}

and is generated by {\mathfrak{n}_1}. Example: Heisenberg group.

Fix a norm on {\mathfrak{n}_1}. Left translates of {\mathfrak{n}_1} define a sub-Riemannian metric, for which dilations {\delta_t=t^i} on {\mathfrak{n}_i} are homothetic.

According to Le Donne, sub-Riemannian Carnot groups are characterized by being the only locally compact homogeneous geodesic metric spaces with homothetic homeos.

Carnot groups come with a left-invariant horizontal basis, we pick a connection on the tangent bundle which makes it parallel. It has torsion. We combine it with the principal bundle connection to define iterated covariant derivatives of curvature. We organize them according to weights adapted to the Lie algebra grading. The above theorem extends.

2.3. Horizontal holonomy

Since we are interested only in holonomy along horizontal loops, we have the freedon to change the connection outside the horizontal subbundle.

Chitour-Grong-Jean-Kokkonen: using this freedom, there are choices which minimize the curvature in the sense that as many components as possible vanish identically. This tends to increase the weight of curvature.

Example: on 3-dimensional Heisenberg group, the preferred connection has curvature which vanishes on the horizontal distribution, hence has weight {\geq 3} instead of 2. Above Theorem provides a 6-th order expansion, whose terms can be computed algebraically.

More generally, on free {k}-step nilpotent Lie groups, the curvature of a preferred connection has order at least {k+1}, whence a {2k+2}-th order expansion whose terms are linear in curvature (in fact, in the preferred curvature).

We expect to use it to refine the Euclidean expansion.

3. Question

What does this give in case of the two-joints snake? Requires to push computations further.

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Notes of Karen Vogtmann’s second Cambridge lecture 23-06-2017

The borders of Outer Space

Joint work with Kai-Uwe Bux and Peter Smillie.

1. Duality groups

I am interested in Poincare duality. For a group, assume {M=B\pi} is a smooth {d}-manifold, then

\displaystyle  \begin{array}{rcl}  H^k_c(\tilde M)=H_{d-k}(\tilde M). \end{array}

Bieri-Eckmann observed that is suffices that {\Gamma} acts freely cocompactly on a contractible space {X} whose compactly supported cohomology vanishes in all degrees but {d}, and {H_c^d(X)} is torsion free. Then {\Gamma} is a duality group.

If the action is merely proper and cocompact, {\Gamma} is a virtual duality group. Borel-Serre used this for lattices. Bestvina-Feighn used this to show that $latex {Out(F_n) is a virtual duality group. Mapping class groups also act on a contractible space.

To achieve cocompactness, Borel-Serre added to the symmetric space copies of Euclidean space forming a rational Euclidean building. The resulting bordification of the quotient is a manifold, with boundary homotopy equivalent to a wedge of spheres (Solomon-Tits). Instead, Grayson constructed an invariant cocompact subset of symmetric space.

Grayson’s work was used by Bartels-Lueck-Reich-Ruping to prove Farrel-Jones for }&fg=000000$Sl(n,{\mathbb Z})$latex {.

For }&fg=000000$Out(F_n){, Bestvina-Feighn defined a bordification too. We proceed differently, like Grayson: we produce and invariant retract }J_n\subset X_n$latex {. It is much easier and gives more information on the boundary.

\section{Construction}

}&fg=000000$X_n{ is the space of metric graphs (without separating edges) with homotopy markings. It is made of simplices with edge-lengths as coordinates. }J_n$ is obtained by chopping off some of their corners.

A core graph is a subgraph such that, when one shrinks it, one gets out of Outer Space. Only corner facing core graphs need be chopped off.

The boundary appears as a union of contractible walls (every intersection of walls is contractible).

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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}

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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<ker(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))}.

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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)<md+d}.

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}.

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