Notes of Pierre-Emmanuel Caprace’s second Cambridge lecture 12-04-2017

Exotic lattices and simple locally compact groups, II

1. The just infinite property

A group is just infinite if it is infinite and all proper quotients are finite.

Examples. {{\mathbb Z}}. Infinite simple groups.

A group is hereditarily just infinite if every finite index subgroup is just infinite.

Example. If {S} is infinite and simple, the wreath product {G=S\wr C_2=(S\times S)\times C_2} where {C_2} swaps the two factors, is ji and not hji.

The Grigorchuk

Theorem 1 (Brenner 1960, Mennicke 1965) For {n\geq 3}, every nontrivial normal subgroup of {G=PSl(n,{\mathbb Z})} contains a congruence subgroup. In particular, {G} is ji.

It exploits the very special properties of this group, and a lot of arithmetic. Since, the theory of arithmetic lattices has split into two branches:

  • Study infinite quotients.
  • Study finite quotients (Congruence Subgroup Problem).

The second branch makes very slow progress due to lack of techniques. Today’s lecture belongs to the first branch.

Theorem 2 (Margulis 1979) Irreducible lattices in centerless semi-simple Lie groups of higher rank are hji.

In rank one, the converse holds: cocompact lattices in rank one are hyperbolic hence SQ-universal. More generally, lattices in hyperbolic locally compact groups are SQ-universal. Indeed, same argument in the cocompact case. In the general case, such lattices are acylindrically hyperbolic groups (Caprace-Cornulier-Monod-Tessera), hence SQ-universal (Dahmani-Guirardel-Osin).

Theorem 3 (Burger-Mozes 2000) Let {\Gamma} be a cocompact lattice in a product of automorphism groups of two leafless trees with at least 3 ends. If the closure of the projection of {\Gamma} to each factor is 2-transitive on the boundary of the tree. Then {\Gamma} is hji.

The main result I will explain today is

Theorem 4 (Bader-Shalom 2006) Let {\Gamma} be a cocompact lattice in a product of locally compact, compactly generated groups {G_1} and {G_2}. Assume that

  1. {G_1} and {G_2} are not both discrete.
  2. {G_1} and {G_2} are just non-compact.
  3. {G_1} and {G_2} do not have normal subgroups isomorphic to {{\mathbb R}^n}.
  4. The projection of {\Gamma} to each factor is dense.

Then {\Gamma} is ji. If furthermore all open finite index subgroups of {G_i} are just non-compact. Then {\Gamma} is hji.

It is a striking theorem, but not so many examples exist, apart from those coveed by theorems 2 and 3. Theorem 4 implies Theorem 3, since 2-transitivity implies just non-compactness.

2. Necessity of assumptions

2.1. Discrete ambient groups

If {G_1} and {G_2} are discrete, {\Gamma} has finite index in the quotient, hence it is virtually a product. Both projections yield proper infinite quotients.

2.2. Non compactly generated groups

Take {G_1={\mathbb Z}} and {G_2=P\Gamma L(2,\bar{\mathbb{F}}_p)}, meaning the semi-direct product with the Galois group

\displaystyle  Aut(\bar{\mathbb{F}}_p)\simeq \hat {\mathbb Z}\simeq\prod_{q\,\mathrm{prime}}{\mathbb Z}_q.

Then {G_2} is just non-compact. Then {\Gamma=P\Gamma L(2,\bar{\mathbb{F}}_p)\times{\mathbb Z}} (where {{\mathbb Z}} acts by the Frobenius automorphism) is a cocompact lattice in {G_1\times G_2}. {G_1} and {G_2} satisfy all assumptions in Theorem 4, except that {G_2} is not compactly generated.

2.3. Just non-compactness

Given a lattice {\Gamma<G_1\times G_2} satisfy all assumptions of Theorem 4. Consider {\Gamma\times\Gamma} is cocompact in {(G_1\times G_1)\times (G_2\times G_2)} which satisfies all assumptions except that {G_1\times G_1} is not just non-compact.

2.4. {{\mathbb R}^n} normal subgroups

{\Gamma={\mathbb Z}^2\times C_2} where {C_2} acts linearly by {\pm 1}, {G_1=G_2={\mathbb R}\times C_2}. View {\Gamma\simeq{\mathbb Z}[\sqrt{2}]\times C_2\subset G_1\times G_2} via the two real embeddings of {{\mathbb Q}[\sqrt{2}]}.

2.5. Non-dense projections

The group {\Gamma_2} introduced last time has non-dense projections. It is not ji.

3. Reduction to topologically simple groups

Theorem 5 (Caprace-Monod) Let {G} be a compactly generated just non compact group. Then one of following holds. Either

  1. {G} is discrete.
  2. {G} is {{\mathbb R}^n} by compact.
  3. {G} is {S\times\cdots\times S} by compact, where {S} is compactly generated, topologically simple, non-compact, non-dicrete.

In addition, in case 3, every non-trivial closed normal subgroup of {G} contains {S\times\cdots\times S}, hence is non-discrete. Any continuous homomorphism {G\rightarrow SU(n)} has finite image.

4. Proof of Theorem 4

Lemma 6 Let {\Gamma} satisfy all assumptions of Theorem 4. Then

  1. Both {G_1} and {G_2} are non-discrete.
  2. Both projections are injective.

Otherwise, assume {G_2} is discrete and {G_1} is non-discrete. The intersection of {\Gamma} with {G_1} is normal in {\Gamma}, hence its projection is normal in {G_1}, so its trivial. Thus projection to {G_2} is injective.

4.1. Strategy

In the remainder, the strategy is due to Margulis. Show that {\Gamma/N} is Kazhdan (Shalom 2000). Show that {\Gamma/N} is amenable (Bader-Shalom 2006, it does not need all assumptions). We focus on the property (T) part.

Definition 7 (Shalom) A group action on a metric space is uniform if there exists {\epsilon} such that every point is moved at least {\epsilon} by some element of a fixed compact subset of the group.

Example. For an orthogonal action, no almost invariant vector means action is uniform on the unit sphere.

If {X} is a proper {CAT(0)} space and {G} acts isometrically on {X} without fixed point either on {X} or on {\partial X}, then action on {X} is uniform.

If {X} is a Hilbert space and {G} acts by affine isometries without fixed points, and if the linear part has no almost invariant vectors, then the actions on {X} is uniform.

4.2. Proof of Property (T)

The proof of property (T) consists of the following 3 results.

Proposition 8 Let {G} be a compactly generated locally compact group which is not Kazhdan. Then there is an affine isometric action on some Hilbert space which is uniform and whose linear part is irreducible.

Proposition 9 (Superrigidity) Let {\Gamma<G_1\times G_2} be a cocompact lattice with dense projections. Assume {G_1} and {G_2} are compactly generated. Let {\Gamma} act uniformly on a Hilbert space {H}. Then the linear part of the action has a sub-representation which extends continuously to {G_1\times G_2} via a projection onto one factor.

Proposition 10 Let {\Gamma<G_1\times G_2} be as in Theorem 4. Then every homomorphism {\Gamma\rightarrow{\mathbb C}} extends to a continuous homomorphism {G_1\times G_2\rightarrow{\mathbb C}}.

Let {N} be a normal subgroup of {\Gamma}. Assume {\Gamma/N} is not Kazhdan. By Proposition 8, get a uniform affine action of {\Gamma/N} on a Hilbert space {H} whose linear part is irreducible. Proposition 9 implies that the linear part {\pi} extends to {\tilde \pi:G_1\times G_2\rightarrow U(H)}. By assumption, {1=\pi(N)=\tilde\pi (proj_1(N))=\tilde \pi(\overline{proj_1(N)})}, hence {\tilde\pi(G_1)} is compact. Since {\pi} was irreducible, so is {\tilde\pi}, so {\tilde\pi} is finite dimensional. By Theorem 5, {\tilde\pi(G_1)} is a finite group, hence {\pi(\Gamma)} is finite. A finite index subgroup of {\Gamma} acts by translation on {H}, whence a non-trivial character. By Proposition 10, {G_1\times G_2} has a character, contradiction.

4.3. Proof of Proposition 9

Here is the key step.

Let {G_1\times G_2} act by isometries on a Hilbert space {H}. Then the linear part has a non-zero invariant vector under {G_1} or {G_2}. (This is an infinite dimensional analogue of a theorem of Monod for actions on proper {CAT(0)} spaces.)

In the setting of Proposition 9, the {\Gamma} action on {H} can be induced to a {G_1\times G_2} on an other Hilbert space, which is still uniform. Its linear part is {Ind_\Gamma^{G_1\times G_2}}. It has non-zero invariant vectors under {G_1} or {G_2}. An elementary Lemma allows to conclude.

Newt week, a new subject: separability and not-residually finite groups.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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