Notes of Alan Reid’s third Oxford lecture 24-03-2017

Profinite rigidity in low dimensions, III

1. Profinite rigidity of classes of closed 3-manifolds

We are interested in the restricted genus

\displaystyle  \begin{array}{rcl}  \mathcal{G}_3(\Gamma)=\{\Delta\, 3-\textrm{compact manifold group}\,\hat\Delta=\hat\Gamma\}. \end{array}

Theorem 1 (Wilton-Zalesskii) Let {\Gamma}, {\Delta} be the fundamental groups of closed 3-manifolds, with {\hat\Delta=\hat\Gamma}.

  1. If one is hyperbolic, so is the other.
  2. If one is a Seifert fibered space, so is the other.

For (1), the point is to detect {{\mathbb Z}\oplus{\mathbb Z}}‘s and free products. Free product decompositions are detected by the first {\ell^2}-Betti number. For {{\mathbb Z}\oplus{\mathbb Z}}‘s, the argument relies on Agol-Wise theory.

For (2), note that there exist non-diffeomorphic Seifert fiber spaces with isomorphic profinite completions.

Garret Wilkes recently proved that Hempel’s is the only possible construction.

Funar showed the existence of non-diffeomorphic Sol manifolds with isomorphic profinite completions. They correspond to explicit matrices which are not conjugate in {Sl_2({\mathbb Z})} but are conjugate in any congruence quotient.

1.1. Rigidity for fibering 3-manifolds

There is a link complement which surjects onto all finite simple groups, and has the same collection of lower central series quotients as {F_2}.

Theorem 2 (Bridson-Reid-Wilton) Let {\Gamma}, {\Delta} be the fundamental groups of finite volume hyperbolic 3-manifolds, with {\hat\Delta=\hat\Gamma}.

  1. If one is fibered with {b_1=1}, so is the other, with the same genus fiber (closed case) or rank of fiber free group (noncompact case).
  2. If {\Gamma} is a 1-punctured torus bundle, then {\mathcal{G}_3(\Gamma)=\{\Gamma\}}.

Boileau-Friedl had a special case of (2), for the figure 8 knot complement.

1.2. Aside on LERF

Suppose {H<\Gamma}. We get a topology induced from the profinite topology of {\hat\Gamma}, and can consider completion {\bar H}, with a continuous surjection {\hat H\rightarrow \bar H}. When is this map injective ?

Example. Answer is positive for finite index subgroups.

Exercise. Say {\Gamma} is {H}-separable if {H} is closed in the profinite topology on {\Gamma}. I.e. {H} is the intersection of finite index subgroups of {\Gamma} that contain it. Say {\Gamma} is LERF if {\Gamma} is {H}-separable for all finitely generated subgroups {H<\Gamma}. Show that above question has a positive answer if {\Gamma} is LERF.

Theorem 3 (Agol, Wise, Scott for Seifert fiber spaces)

  1. Geometric closed 3-manifold groups are LERF.
  2. Non-compact finite volume hyperbolic manifold groups and non-compact Seifert fibered space groups are LERF.

The converse of (1) is true (Hongbin Sun).

Non-example. For {\Gamma=Sl_3({\mathbb Z})} and {H} be a corner {Sl_2({\mathbb Z})}, answer is no. This follows from the Congruence subgroup property.

2. Proofs

Once fibering is established, the genus and rank statement follows easily. Indeed, thanks to LERF, exact sequences yield exact sequences of profinite completions. {b_1=1} makes the exact sequence unique, so the kernel is uniquely defined.

For 1-punctured torus bundles, there are only 2 possibilities for the fiber. One shows that a triply punctured sphere fiber arises only for Seifert fibered spaces. Hyperbolic 1-punctured torus bundles have monodromy a hyperbolic element {\phi} of {Sl_2({\mathbb Z})} (Jorgensen). The cardinality of the torsion part of {H_1} is {|trace(\phi)-2|}, so this leaves only finitely many possibilities.

2.1. Proof of BRW

Assume that {M} is finite volume non-compact hyperbolic. Let {N} be profinite equivalent to {M}. Start with a few reductions.

Assume {N} is compact with a single incompressible torus boundary component. Task: show that {N} is fibered.

Assume that {N} is finite volume hyperbolic. {b_1=1} implies that {\Delta} surjects onto {{\mathbb Z}} with kernel {K}. Freedman shows that {K} contains a closed surface group {H}. Wise shows that {\Delta} is LERF, so exact sequences go to profinite completions.

Beware that if {F} is a free group, {\hat F} contains dense surface groups (Breuillard-Gelander-Souto-Storm). So one must take care of closedness of subgroups. There is a notion of cohomological dimension for profinite groups. Closed surface subgroups contribute, but not dense surface subgroups.

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Notes of Enrico Le Donne’s Oxford lecture 24-03-2017

From homogeneous spaces to Lie groups

With Michael Cowling, Ville Kivioja, Alessandro Ottazi and Sebastiano Nicolussi Golo.

1. Homogeneous metric spaces

I.e. with a transitive group of isometries.

Solenoids (inverse limits of {S^1} with maps {t\mapsto 2t}) are homogeneous. {p}-adics are too. We shall stick to connected metric spaces.

From the solution to Hilbert’s 5th problem, it follows that a connected, locally connected and locally compact homogeneous space is {G/H} with {G} a Lie group and {H} a compact group. Call this a Lie homogeneous space.

It follows that if {X} is a connected and locally compact homogeneous space, then for all {\epsilon>0}, {X} is {(1,\epsilon)}-quasi-isometric to a metric Lie group (choose the metric on the isometry group {G} in order that {H} has small diameter). Up to passing to a subgroup, one can assume {G} to be solvable.

2. Isometric metric Lie groups

Theorem 1 Given a metric Lie group {(G,d_G)}, there is a {(1,\epsilon)}-qi invariant metric {d'_G} which is isometric to a metric Lie group of the form {K\times S} where {K} is compact and {S} is solvable.

Example. {\widetilde{Sl_2}} can be made isometric to {\mathbb{H}^2 \times{\mathbb R}}.

3. The polynomial growth case

For a general (non-geodesic) metric Lie group, polynomial growth does not imply doubling. Nevertheless,

Theorem 2 For a metric Lie group {(G,d_G)} of polynomial growth, there is a {(1,\epsilon)}-qi invariant metric {d'_G} which is isometric to a metric Lie group of the form {K\times N} where {K} is compact and {N} is nilpotent.

{N} is uniquely defined, it is the nilshadow of the solvable group {S}.

In fact, the following can be extracted from work of Gordon and Wilson. Given simply connected Lie groups {H} and {N}, with {N} nilpotent, then {H} and {N} can be made isometric iff {H} is solvable of type (R) and {N} is its nilshadow.

4. Playing with distances

On {{\mathbb R}^n}, there is a wealth of left-invariant distances (snowfloakes, bounded,…). Idem for nilpotent groups. Geodesicity, existence of one dilation suffices to characterize Carnot groups with sub-Finsler metrics.

What if geodesicity is relaxed?

We consider gradings on Lie algebras indexed by real numbers {\geq 1}. Such algebras are automatically nilpotent.

Theorem 3 (Hebisch-Sikora) Every simply connected Lie group whose Lie algebra is graded admits an invariant distance which is dilation invariant.

Example. Heisenberg with 3 homogeneous components.

Theorem 4 (Hebisch-Sikora) If {X} is a metric space which is locally compact, connected homogeneous and admit one dilation. Then {X} is a graded Lie group with a left-invariant metric and homogeneous under dilations.

This a combination of many existing results. Since {X} is doubling, {ISo(X)} is Lie with polynomial growth. The Killing orthogonal to a stabilizer exponentiates into a subgroup with a simply transitive action on {X}. So {X} is a metric Lie group.

5. Quiz

Say finitely generated groups are equivalent if they admit isometric word metrics. Is this an equivalence relation?

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Notes of Emmanuel Breuillard’s third Oxford lecture 24-03-2017

Approximate groups, III

1. Growth

Let {F} be a finitely generated free group. It is easy that for the standard generating system, {|S|^n=2k(2k-1)^{n-1}}. What about other generating systems?

Theorem 1 (Razborov, Safin) For an arbitrary generating set {S} in {F},

\displaystyle \begin{array}{rcl} |S^{2n-1}|\geq(\frac{|S|}{10})^n . \end{array}

 

If a Product Theorem {|AAA\geq |A|^{1+\epsilon}} holds in a group, then {|A^n|\geq |A|^{n^\alpha}}. So this implies super-polynomial growth, but not exponential growth. Nevertheless, it might give an alternative proof of (a weak form) of Gromov’s polynomial growth theorem. Note that Product Theorems in this form are known only for certain finite simple groups (and even fail for alternating groups).

1.1. Proof of Gromov’s polynomial growth theorem

Indeed, the structure theorem for approximate groups (BGT) yields such a proof.

Assume that for all {n}, {|S^n|\leq C\,n^d}. There are arbitrarily large integers such hat {|S^{2r}|\leq 3^d|S^r|}, i.e. {S^r} is an approximate subgroup.

BGT states that {S^r\subset X_r H_r} where {H_r} is virtually nilpotent and {|X_r|\leq const.(d)}. For {r>|X_r|}, this shows that {H_r} has finite index.

Remark. We merely need one large value of {r} such that {|S^{2r}|\leq 3^d|S^r|}. Also, the constant {C} plays no role, merely the ratio {|S^r|/|S|}. So that applies to generating sets of arbitrary size.

2. Margulis Lemma

Margulis Lemma states that in a compact negatively curved Riemannian manifold {M}, the subgroup of {\pi_1(M,x)} generated by loops of length {<\epsilon(}dimension,min sectional curvature{)} is cyclic. In the non-compact case, cyclic needs be replaced with nilpotent.

Here is a generalization, which follows from BGT.

Lemma 2 Let {X} be a metric space with {K}-bounded geometry (every ball of radius 4 is covered by at mots {K} balls of radius 1). Then, for every {x}, any discrete group of isometries generated by elements that move {x} at most {\epsilon(K)} away is virtually nilpotent.

Proof. Let

\displaystyle \begin{array}{rcl} S_\epsilon=\{\gamma\,;\, d(x,\gamma)<\epsilon\}. \end{array}

Then {S_2} is an approximate group, hence contained in {XH} with {H} virtually nilpotent and {|X|\leq const.(K)}, {H} has finite index.

3. Proof of structure theorem

It goes by contradiction. Compactness plays a role, in the form of ultraproducts. The strategy was outlined by Hrushovski. We dug into the proof of the solution to Hilbert’s 5th problem.

Take any sequence of approximate subgroups {A_n} in groups {G_n}. Form ultraproducts {A<G}. This is a non-standard approximate subgroup.

3.1. Step 1

Define a locally compact topology on {A}.

A lemma due to independently to Sanders and Hrushovski says that in a finite {K}-approximate group {A'}, there exists a subset {S} of positive proportion such that {S^k\subset A'^4}. Such sets provide a basis for the required topology.

Then our non-standard {A} maps to a locally compact group with image a compact neighborhood of {1}.

3.2. Step 2

Use tools from the proof of the solution to Hilbert’s 5th problem: up to passing to an open subgroup, can mod out by a compact group and get a Lie group.

The point (Gleason Lemmas) is to show that one-parameter subgroups (obtained as limits of cyclic subgroups) can be multiplied. This produces a vectorspace, candidate to be the Lie algebra. These lemmas are quantitative, we can give approximative group versions.

4. Questions

Does this give a gap from polynomial growth? Not quite, because non effective. Kleiner’s proof is effective, Shalom and Tao managed to extract a gap from it, something like {n^{\log\log n}}.

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Notes of Bruno Duchesne’s Oxford lecture 23-03-2017

Groups acting on dendrites

Joint with Nicolas Monod.

1. Dendrites

A dendrite is a continuum (compact metrizable connected spaces) which is locally connected and in which every two points are joined by a unique arc (continuous injective image of an interval).

Example. The end compactification of a tree is a dendrite.

Every dendrite can be metrized to become an {{\mathbb R}}-tree. Conversely, for every second countable {{\mathbb R}}-tree, there is a weaker topology (observer’s topology, i.e. the topology than makes point complements open) that turns it s end compactification into a dendrite.

Example. Start with a finite tripod. Glue a smaller tripod at the middle of each edge. Iterate. Get the Wasewski dendrite {D_\infty}, homeomorphic to Berkovich’ projective line over {C_p}.

2. Actions on dendrites

In view of the proximity of trees and dendrites, it is tempting to ask

Question. Can a Kazhdan group act non-elementarily on a dendrite ?

Elementary means that there is either a fixed point or an invariant arc.

2.1. Invariant measures

The order of a dendrite at a point {x} is the number of connected components of the complement. It is at most countable. We speak of

  • ends, with order 1,
  • regular points, with order 2,
  • branch points, with order {\geq 3}.

The set of branched points is at most countable,

Proposition 1 For an action of {G} on a dentrite {X}, the following are equivalent.

  1. Action is elementary.
  2. There is a finite orbit.
  3. There is an invariant probability measure.

 

It follows that actions of amenable groups on dendrites are elementary.

2.2. Minimality

Dendro-minimality means that no proper invariant subdendrites (i.e. closed connected subspaces).

On shows that a non-elementary action contains a unique closed minimal invariant subset which is contained in a unique minimal invariant subset which is a subdendrite.

Theorem 2 (Tits alternative) Either that action is elementary or {G} contains a free group.

Proof. First reduce to minimal action. Then find elements with a north-south dynamics and play ping-pong.

Theorem 3 For every non-elementary action there is a canonical unitary representation {\pi} of {G} with a nonzero element {w\in H^2_b(G,\pi)}.

It follows that higher rank lattices do not act non-elementarily on dendrites.

3. Back to Wazewski dendrites

Here is a characterization. Given a subset {S\subset \{3,4,\ldots}, {D_S} is the unique dendrite such that

  1. Order of all brache points belong to {S},
  2. For all {n\in S}, the set of order {n} branch points of {D_S} is arcwise dense.

Note that for every connected open subset {O} of {D_S}, {\bar O} is homeomorphic to {D_S}.

Any finite subset {F\subset D_S} defines a dendrite {[F]} which is a finite tree. It inherits the structure of a finite labelled graph (labels are orders of endpoints).

Proposition 4 Two finite subsets of {D_S} defining isomorphic labelled graphs are equivalent by a homeomorphism.

It follows that, for finite {S}, the action of {Homeo(D_S)} on {D_S} is oligomorphic, i.e. it has finitely many orbits on {n}-tuples for all {n}.

Theorem 5 {Homeo(D_S)} is simple. {Homeo(D_S)\simeq Homeo(D_{S'})} iff {S=S'}.

Indeed, the stabilizer of a point {x} is a wreath product the homeomorphism groups of branches (components of complement of {x}).

3.1. Topologies on {Homeo(D_S)}

Uniform convergence on {D_S} and pointwise convergence on branch points coincide. Hence it is Polish.

Theorem 6 If {S} is finite, {Homeo(D_S)} is Kazhdan. Furthermore, it admits a finite Kazhdan set.

Say a group has property (OB) if in any isometric action on a metric space, orbits are bounded.

The prototype is {Homeo(S^1)}. For locally compact groups, this can happen only if group is compact.

Theorem 7 {Homeo(D_S)} has property (OB).

It follows that {Homeo(D_S)} has property (FH) and property (FA). Note that, since {Homeo(D_S)} is not locally compact, (FH) need not imply (T). The proof that {Homeo(D_S)} is Kazhdan follows a different route.

3.2. Proof of property (T)

If {S} is finite, {Homeo(D_S)} is oligomorphic and simple. According to Evans and Tsankov, this implies property (T) with a finite Kazhdan set. Indeed, one shows that every unitary representation is a direct sum of irreducibles. Every irreducible representation is of the following form: given a finite subset {F\subset D_S}, consider

\displaystyle \begin{array}{rcl} \pi_S = \ell^2(Homeo(D_S)/Stab(F)). \end{array}

For such representations, property (T) is proven by hand.

3.3. Proof of property (OB)

By transitivity on pairs of endpoints, {Homeo(D_S)\subset Stab(x)\cdot Stab(y)}. It suffices to show that {Stab(x)} has property (OB). Since {x} is fixed, there is a partial order on {D_S}. It is semi-linear and dense. The action on branched points is weakly transitive. The automorphism group of this order lifts to {D_S}. According to Droste-Truss, this group has (OB), hence {Homeo(D_S)} has (OB).

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Notes of Marc Burger’s third Oxford lecture 23-03-2017

Geometric structures, compactifications of representation varieties, and non archimedean geometry, III

1. To infinity (and beyond)

1.1. Compactification

We intend to describe the boundary of the representation variety {Rep_{max}(\Gamma,Sp(2n,{\mathbb R}))}.

Theorem 1 (Burger-Pozzetti) If {\rho:\Gamma\rightarrow Sp(2n,{\mathbb R})} is a maximal representation, then

  1. Nontrivial elements do not have eigenvalues of modulus 1. In particular, translation lengths are {>0}.
  2. If {\gamma} and {\eta} represent intersecting closed geodesics in {\Sigma}, then the following analogue of Collar Lemma holds,

    \displaystyle  \begin{array}{rcl}  (e^{\frac{\ell(\rho(\gamma))}{\sqrt{n}}}-1)(e^{\frac{\ell(\rho(\eta))}{\sqrt{n}}}-1)\geq 1. \end{array}

In particular, the vector-valued length function {\nu_\rho} never vanishes.

Corollary 2 (Parreau) The projectived length map

\displaystyle  \begin{array}{rcl}  \mathbb{P}\circ\nu:Rep_{max}(\Gamma,Sp(2n,{\mathbb R}))\rightarrow \mathbb{P}({\mathfrak{a}^+}^\Gamma) \end{array}

has relatively compact image and any boundary point is the vector-valued length function of a {\Gamma}-action on a building associated to {Sp(2n,{}^{\omega}{\mathbb R}_\lambda)} where {{}^{\omega}{\mathbb R}_\lambda} is a Robinson field.

There is a loss of information.

Question 1. Does this building have special geometric properties?

Question 2. Is there a way to organize all these actions on buildings into a coherent compactification of {Rep_{max}(\Gamma,Sp(2n,{\mathbb R}))}?

1.2. Answer to question 1

The following is known since the 1980’s.

Theorem 3 (Skora) If {n=1}, boundary points are exactly length functions of actions on {{\mathbb R}}-trees with small stabilizers, i.e. stabilizers of germs of segments are either trivial or cyclic.

For {n\geq 2}, we need study sequences {(\rho_k)} in {Hom_{max}(\Gamma,Sp(2n,{\mathbb R}))} and the resulting actions an asymptotic cones of {\mathcal{X}_n}.

A sequence leaves to infinity if some marked point, say {o=iId\in\mathcal{X}_n}, is moved farther and farther away by some element of a fixed generating system {S}. Pick a sequence {\lambda=(\lambda_k)_{k\geq 1}} such that

\displaystyle  \begin{array}{rcl}  \max_{s\in S}d(\rho_k(s)o,o)=O(\lambda_k). \end{array}

Pick a non-principal ultrafilter {\omega}. Form the corresponding asumptotic cone {{}^\omega\mathcal{X}_\lambda}. It is a complete {CAT(0)} metric space, with an isometric action {{}^\omega\rho_\lambda} of {\Gamma}.

Example. {n=1}. Assume {\rho_k} pinches some closed geodesic {\gamma}, and does not affect curves disjoint from {\gamma}. Due to the Collar Lemma, any closed curve intersecting {\gamma} has length tending to infinity at speed {\lambda_k=\log(1/}length{\rho_k(\gamma))}. In the limit, {{}^\omega\rho_\lambda(\gamma)} has a fixed point.

Definition 4 A simple closed geodesic {c} on {\Sigma} is special if

  1. {\ell({}^\omega\rho_\lambda(\gamma))=0} whenever {\gamma} represents {c}.
  2. For any closed geodesic {c'} intersecting {c}, {\ell({}^\omega\rho_\lambda(\gamma))>0} whenever {\eta} represents {c'}.

There are at most {3g-3} special geodesics.

Theorem 5 The isometric action {{}^\omega\rho_\lambda} is faithful. Let {\Sigma_v} be a component of the complement in {\Sigma} of special geodesics. There is a dichotomy:

  1. Either (PT): every curve in {\Sigma_v} which is not a boundary component has {\ell({}^\omega\rho_\lambda(c))>0}.
  2. Or (FP): {\pi_1(\Sigma_v)} fixes a point in {{}^\omega\mathcal{X}_\lambda}.

This is used in the proof of

Theorem 6 (Burger-Pozzetti-Iozzi-Parreau) The {\Gamma} action {{}^\omega\rho_\lambda} on {{}^\omega\mathcal{X}_\lambda} is small.

An other tool is the theory of maximal representations of surfaces with boundary. Indeed, the scale {\lambda} is too large for certain components, thus a scale needs be chosen for each component. Only finitely many scales arise.

Personnally, I am not too fluent with buildings, I prefer the language of Robinson fields.

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Notes of Alireza Salehi Golsefidy’s Oxford lecture 23-03-2017

Super-approximation

1. Definition

Say a finitely generated subgroup {\Gamma} of {Gl_n({\mathbb Q})} (with fixed generating set {\Omega}) has super-approximation with respect to a set {C} of positive integers which are coprime to {q_0} if the family of Cayley graphs {Cay(\pi_q(\Gamma),\pi_q(\Omega))} is an expander. Here,

\displaystyle  \begin{array}{rcl}  \pi_q:{\mathbb Z}[\frac{1}{q_0}]\rightarrow{\mathbb Z}[\frac{1}{q_0}]/q{\mathbb Z}[\frac{1}{q_0}]. \end{array}

Example. When {C} is the set of powers of a single prime {p}, the inverse limite of finite groups {\pi_{p^n}} is the closure of {\Gamma} in {Gl_n({\mathbb Z}_p)}, {{\mathbb Z}_p} equals {p}-adic integers.

Expansion is expressible in terms of convolution of measures. Let {\mu} denote the uniform probability measure on given generating set {\Omega}. Let {G} be a compact group containing {\Gamma}. Let {T} denote the averaging operator

Definition 1 Let {\lambda(\Omega;G)} denote the operator norm of {T} on the orthogonal complement of constant functions in {L^2} of the closure of {\Gamma},

\displaystyle  \begin{array}{rcl}  \lambda(\Omega;G)=\|T\|_{|L^2_0(\bar\Gamma}\| \end{array}

Then super-approximation is equivalent to

\displaystyle  \begin{array}{rcl}  \sup_{q\in C}\lambda(\Omega,Gl_n({\mathbb Z}/q{\mathbb Z}))<1. \end{array}

Example. When {C} is the set of powers of a single prime {p}, super-approximation is equivalent to

\displaystyle  \begin{array}{rcl}  \lambda(\Omega,Gl_n({\mathbb Z}_p))<1. \end{array}

2. Results

Follow from work of many people.

2.1. Arithmetic lattices

Theorem 2 Let {G} be a semi-simple {{\mathbb Q}}-group, then {\Gamma=G({\mathbb Z}[\frac{1}{q_0}])} has super-approximation (if it is infinite).

Selberg showed this for {Sl_2}. Burger-Sarnak showed that this property passes from lattices in one group to another. Jacquet-Langlands used these to handle real rank one groups, with the exception of unitary groups, solved by Clozel. Clozel-Ullmo did the {p}-adic case. In higher rank, Kazhdan-Margulis.

2.2. From arithmetic to more general groups

Theorem 3 (Bourgain-Varju) Let {\Gamma<SL_n({\mathbb Z})} be Zariski-dense in {Sl_n}. Then {\Gamma} has super-approximation.

This relies on the dynamics of {Sl_n({\mathbb Z})} on the torus, classification of invariant measures. The method is limited to archimedean fields.

Theorem 4 (Salehi Golsefidy-Varju) Let {\Gamma<G({\mathbb R})} have Zariski-closure {G^0}. Then {\Gamma} has super-approximation with respect to {N_0}-th powers of all square-free integers coprime to {q_0} iff {G^0=[G^0,G^0]}.

Infinite abelianization easily make spectral gap impossible. It is the converse which is hard.

Theorem 5 (Salehi Golsefidy) Let {\Gamma<G({\mathbb R})} have Zariski-closure {G^0}. Then {\Gamma} has super-approximation with respect to all powers of all primes coprime to {q_0} iff {G^0=[G^0,G^0]}.

Theorem 6 (Salehi Golsefidy-Zhang) Let {\Lambda<\Gamma<Gl_n({\mathbb Z}[\frac{1}{q_0}])} have Zariski-closures {H} and {G}. Assume that {G^0} is the smallest normal subgroup of {G^0} containing {H^0}. Then Then {\Gamma} has super-approximation if {\Lambda} does.

This provides new examples, among subgroups of arithmetic lattices.

3. Applications

Strong approximation describes the closure of {\Gamma} in compact groups. Combined with super-approximation, this leads to interesting results.

3.1. Affine sieve

Theorem 7 (Salehi Golsefidy-Sarnak) Let {\Gamma<Gl_n({\mathbb Z}[\frac{1}{q_0}])}, with Zariski closure {G}, {G^0=[G^0,G^0]}. Let {f} be a rational polynomial that does not vanish identically on {G^0}. There exist integers {r} and {q'_0} such that

\displaystyle  \begin{array}{rcl}  \Gamma_{r,q'_0}(f)=\{\gamma\in\Gamma\,;\,f(\gamma)=p_1\cdots p_{r'},\,r'\leq r,\,p_i\textrm{ primes in }{\mathbb Z}[\frac{1}{q_0}]\} \end{array}

is Zariski-dense in {G}.

This is in the spirit of Dirichlet’s theorem on primes in arithmetic progressions: we produce elements with few prime factors in the set of values of some polynomial on a subgroup. A special case is Oh-Kantorovitch’s work on inverse radii of Appolonian circles.

3.2. Sieve in groups

Theorem 8 (Lubotzky-Meiri) A non virtually solvable finitely generated subgroup {\Gamma} of {Gl_n(F)} ({F} of characteristic 0) is not covered by finitely many shifts of its powers {\bigcup_{m\geq 2}\Gamma^m}.

3.3. Orbit-equivalence rigidity

Let {\Gamma<G} be a dense subgroup in a {p}-adic analytic semisimple Lie group {G}. Assume {Ad(\Gamma)} is represented by matrics with algebraic entries in some basis. Let {H} be a locally profinite group, and {\Lambda<H} a dense countable subgroup. Then the actions of {\Gamma} on {G} and of {\Lambda} on {H} are measurably orbit equvalent, then there is an isomorphism between open subgroups of {G} and {H} which maps (the intersections of) {\Gamma} to {\Lambda}.

Question. Does this imply that {\Gamma} and {\Lambda} are isomorphic ?

3.4. Deformations of Galois representations

Ellenberg-Hall-Kowalski use super-approximation to give a new proof of a result of Cadoret-Tamagawa on abelian schemes over curves.

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Notes of Alan Reid’s second Oxford lecture 23-03-2017

Profinite rigidity in low dimensions, II

1. Profinite completion

We organize finite quotients of a group {\Gamma}. The set {\mathcal{N}} of normal finite index subgroups of {\Gamma} is a directed set, where {N_1\langle N_2} if {N_1} contains {N_2} (unfortunate conflict of notation).

The profinite completion {\hat\Gamma} of {\Gamma} is the inverse limite of finite quotients {\Gamma/N}, {N\in\mathcal{N}}. It can be viewed as a subset in the direct product {\prod_{N\in\mathcal{N}} \Gamma/N}. It is a compact topological group.

{\Gamma} maps to {\hat\Gamma}, its image is dense. This map is injective iff {\Gamma} is residually finite.

Nikolov-Segal: finite index subgroups of {\hat\Gamma} are open.

Theorem 1 (Correspondence theorem) There is a 1-1 correspondence between subgroups of finite index of {\Gamma} and open subgroups of {\hat\Gamma}. Indices and normality are preserved. It follows that

\displaystyle \begin{array}{rcl} \mathcal{C}(\Gamma)=\mathcal{C}(\hat\Gamma). \end{array}

 

Theorem 2 Let {\Gamma_1} and {\Gamma_2} be finitiely generated residually finite groups. Then {\mathcal{C}(\Gamma_1)=\mathcal{C}(\Gamma_2)} iff {\hat\Gamma_1\simeq\hat\Gamma_2}. Hence {\hat\Gamma} can be recovered from the set of finite quotients of {\Gamma}.

Definition 3 Say {\Gamma} is profinitely rigid if {\mathcal{G}=\{\Gamma\}}.

Conjecturally, free groups, surface groups, finite volume hyperbolic 3-manifold groups are profinitely rigid. Unclear for more general hyperbolic groups.

Not so much is known about the genus. Genus is finite for nilpotent groups. Also often for groups of arithmetic origin. It cost some efforts to find an example of a group with infinite genus (Bridson after Grunewald).

2. Crash course on 3-manifolds

Let {M} be closed, orientable, 3 dimensional. Kneser-Milnor: {M} decomposes as a connected sum of finitely manyprime manifolds (i.e. which cannot we decomposed further). From now on, stick to prime manifolds.

Either {\pi_1} is finite. Then (Perelman) {M} is covered by {S^3}. Otherwise, {\tilde M\simeq{\mathbb R}^3} or {S^2\times {\mathbb R}}. In the latter case, {\pi_1={\mathbb Z}}. From now on, stick to prime manifolds with infinite {\pi_1} different from {{\mathbb Z}}.

{M} either admits a geometric structure or admits a decomposition along incompressible tori. In the latter case, {\pi_1} has a non-trivial JSJ decomposition. In the former case, {M} is modelled on Euclidean space {\mathbb{E}^3}, {\mathbb{H}^3}, {\mathbb{H}^2\times{\mathbb R}}, Nil, Sol, {\widetilde{Sl}_2}. Manifolds modelled on {\mathbb{E}^3}, {\mathbb{H}^2\times{\mathbb R}}, Nil and {\widetilde{Sl}_2} are Seifert fibered spaces. {\mathbb{H}^2\times{\mathbb R}}-manifolds are virtually products surface {\times} circle. Manifolds modelled on {\mathbb{E}^3}, Nil and Sol are virtually torus bundles over the circle. {\widetilde{Sl}_2}-manifolds are never suface bundles.

Do hyperbolic manifolds virtually fiber over the circle? This has been open for decades. Agol came up with an idea which applied to a small class. Then Wise designed the theory of special cube complexes which fitted nicely with Agol’s initial idea. This has resulted into a proof that all hyperbolic manifolds virtually fiber.

Remark. Hempel has shown that there exist Seifert fibered bundles, which are not profinitely rigid.

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