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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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.

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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<G} 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<G} 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<G}, 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<G} 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<G} 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<G} be hyperbolically embedded. Then (EP) is solvable for {H<G}. 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<G}.

  1. If {H} is virtually cyclic, then (EP) is solvable for {H<G}.
  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<G}.
  3. Conversely, if (EP) for {H<G} is solvable, then {H} is quasi-convex.

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Notes of Jean-Fran├žois 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.

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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}.

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