## 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 ${{\mathbb R}^n}$.

## Notes of Matt Clay’s Cambridge lecture 22-06-2017

Homology of curves and surfaces

Question. Given space ${X}$ and a multicurve ${\gamma}$ in ${X}$ which is null-homologous. Does ${\gamma}$ bound an essential surface in ${X}$? Essential means ${\pi_1}$-injective.

Interesting cases are

• Surfaces.
• 3-manifolds,
• Shimura varieties.

1. Surface case

Theorem 1 (Calegari) In a higher genus surface, every null-homologous multicurve virtually bounds an immersed surface.

Virtually means that some multiple ${n\gamma}$ does.

Calegari does it by hand, by homologing to simpler multicurves, with less intersections. The figure eight curve in the plane does not bound any immersed surface, but its double does.

2. Case of hyperbolic 3-manifolds

Say a surface ${S}$ immersed in a 3-manifold ${X}$ is panted if a pants decomposition is provided such that the restriction of the immersion to each pant is essential.

Define the panted cobordism group ${\Omega(X)}$ as the set of formal sums of multicurves, modulo cobounding a panted surface.

Theorem 2 (Liu-Markovic) If ${X}$ is a 3-manifold, there is an isomorphism

$\displaystyle \begin{array}{rcl} \Omega(X)\simeq H_1(SO(X),{\mathbb Z}). \end{array}$

where ${SO(X)}$ is the special orthonormal frame bundle of ${X}$.

Using the homotopy exact sequence of the fibration ${SO(X)\rightarrow X}$, get

Corollary 3 There exists a nontrivial ${{\mathbb Z}/2}$-valued invariant ${\sigma}$ on null-homologous multicurves, which is additive and vanishes on boundaries of panted surfaces.

The idea is that cutting a surface along a pants decomposition and regluing it in a different way does not change homology. In this way, one ultimately gets an essential surface.

The same technology yields the following

Theorem 4 (Liu-Markovic) If ${X}$ is a hyperbolic 3-manifold, every rational second homology class has a positive integral multiple represented by an oriented essential quasi-Fuchsian surface.

Indeed, Dehn’s Lemma provide a Haken surface. String it: use a pants decomposition of it, add break the surface.

Sun has continued:

Theorem 5 (Sun) For any finite abelian group ${A}$, every hyperbolic 3-manifold ${X}$ has a finite cover of which the first integral homology has ${A}$ as a direct summand.

Theorem 6 (Sun) For any closed orientable 3-manifold ${N}$, every hyperbolic 3-manifold ${X}$ has a finite cover that 2-dominates ${N}$, meaning that it has degree 2 map to ${N}$.

Degree 2 has been improved to 1 recently. The number 2 arose from our ${{\mathbb Z}/2}$ obstruction above.

3. Case of Shimura varieties

Let ${G}$ be a product of copies of ${PSl(2,{\mathbb R})}$ and ${\Gamma}$ an irreducible lattice in ${G}$. Every example arises from a totally real number field ${K}$, a quaternion algebra over ${K}$ and a maximal order in it.

We show that there exists a surface subgroup ${F\subset\Gamma}$ whose projections to factors are Fuchsian.

Corollary 7 There exists a nontrivial ${{\mathbb Z}/2}$-valued invariant ${\sigma}$ on null-homologous multicurves, which is additive and vanishes on boundaries of essential surfaces.

Theorem 8 (Kahn-Markovic) Let ${K}$ be a totally real number field ${K}$, ${A}$ a quaternion algebra over ${K}$ such that ${A\otimes_K {\mathbb R}\not=M(2,{\mathbb R})}$. Then there exists a Riemann surface ${S}$ whose trace field and quaternion algebra are equal to ${K}$ and ${A}$. Furthermore, group elements of the surface groups are integers in ${A}$.

## Notes of Vlad Markovic’s second Cambridge lecture 22-06-2017

Homology of curves and surfaces

Question. Given space ${X}$ and a multicurve ${\gamma}$ in ${X}$ which is null-homologous. Does ${\gamma}$ bound an essential surface in ${X}$? Essential means ${\pi_1}$-injective.

Interesting cases are

• Surfaces.
• 3-manifolds,
• Shimura varieties.

1. Surface case

Theorem 1 (Calegari) In a higher genus surface, every null-homologous multicurve virtually bounds an immersed surface.

Virtually means that some multiple ${n\gamma}$ does.

Calegari does it by hand, by homologing to simpler multicurves, with less intersections. The figure eight curve in the plane does not bound any immersed surface, but its double does.

2. Case of hyperbolic 3-manifolds

Say a surface ${S}$ immersed in a 3-manifold ${X}$ is panted if a pants decomposition is provided such that the restriction of the immersion to each pant is essential.

Define the panted cobordism group ${\Omega(X)}$ as the set of formal sums of multicurves, modulo cobounding a panted surface.

Theorem 2 (Liu-Markovic) If ${X}$ is a 3-manifold, there is an isomorphism

$\displaystyle \begin{array}{rcl} \Omega(X)\simeq H_1(SO(X),{\mathbb Z}). \end{array}$

where ${SO(X)}$ is the special orthonormal frame bundle of ${X}$.

Using the homotopy exact sequence of the fibration ${SO(X)\rightarrow X}$, get

Corollary 3 There exists a nontrivial ${{\mathbb Z}/2}$-valued invariant ${\sigma}$ on null-homologous multicurves, which is additive and vanishes on boundaries of panted surfaces.

The idea is that cutting a surface along a pants decomposition and regluing it in a different way does not change homology. In this way, one ultimately gets an essential surface.

The same technology yields the following

Theorem 4 (Liu-Markovic) If ${X}$ is a hyperbolic 3-manifold, every rational second homology class has a positive integral multiple represented by an oriented essential quasi-Fuchsian surface.

Indeed, Dehn’s Lemma provide a Haken surface. String it: use a pants decomposition of it, add break the surface.

Sun has continued:

Theorem 5 (Sun) For any finite abelian group ${A}$, every hyperbolic 3-manifold ${X}$ has a finite cover of which the first integral homology has ${A}$ as a direct summand.

Theorem 6 (Sun) For any closed orientable 3-manifold ${N}$, every hyperbolic 3-manifold ${X}$ has a finite cover that 2-dominates ${N}$, meaning that it has degree 2 map to ${N}$.

Degree 2 has been improved to 1 recently. The number 2 arose from our ${{\mathbb Z}/2}$ obstruction above.

3. Case of Shimura varieties

Let ${G}$ be a product of copies of ${PSl(2,{\mathbb R})}$ and ${\Gamma}$ an irreducible lattice in ${G}$. Every example arises from a totally real number field ${K}$, a quaternion algebra over ${K}$ and a maximal order in it.

We show that there exists a surface subgroup ${F\subset\Gamma}$ whose projections to factors are Fuchsian.

Corollary 7 There exists a nontrivial ${{\mathbb Z}/2}$-valued invariant ${\sigma}$ on null-homologous multicurves, which is additive and vanishes on boundaries of essential surfaces.

Theorem 8 (Kahn-Markovic) Let ${K}$ be a totally real number field ${K}$, ${A}$ a quaternion algebra over ${K}$ such that ${A\otimes_K {\mathbb R}\not=M(2,{\mathbb R})}$. Then there exists a Riemann surface ${S}$ whose trace field and quaternion algebra are equal to ${K}$ and ${A}$. Furthermore, group elements of the surface groups are integers in ${A}$.