** 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**. . Infinite simple groups.

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

**Example**. If is infinite and simple, the wreath product where swaps the two factors, is ji and not hji.

The Grigorchuk

Theorem 1 (Brenner 1960, Mennicke 1965)For , every nontrivial normal subgroup of contains a congruence subgroup. In particular, 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 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 to each factor is 2-transitive on the boundary of the tree. Then is hji.

The main result I will explain today is

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

- and are not both discrete.
- and are just non-compact.
- and do not have normal subgroups isomorphic to .
- The projection of to each factor is dense.

Then is ji. If furthermore all open finite index subgroups of are just non-compact. Then 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 and are discrete, 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 and , meaning the semi-direct product with the Galois group

Then is just non-compact. Then (where acts by the Frobenius automorphism) is a cocompact lattice in . and satisfy all assumptions in Theorem 4, except that is not compactly generated.

** 2.3. Just non-compactness **

Given a lattice satisfy all assumptions of Theorem 4. Consider is cocompact in which satisfies all assumptions except that is not just non-compact.

** 2.4. normal subgroups **

where acts linearly by , . View via the two real embeddings of .

** 2.5. Non-dense projections **

The group introduced last time has non-dense projections. It is not ji.

**3. Reduction to topologically simple groups **

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

- is discrete.
- is by compact.
- is by compact, where is compactly generated, topologically simple, non-compact, non-dicrete.

In addition, in case 3, every non-trivial closed normal subgroup of contains , hence is non-discrete. Any continuous homomorphism has finite image.

**4. Proof of Theorem 4 **

Lemma 6Let satisfy all assumptions of Theorem 4. Then

- Both and are non-discrete.
- Both projections are injective.

Otherwise, assume is discrete and is non-discrete. The intersection of with is normal in , hence its projection is normal in , so its trivial. Thus projection to is injective.

** 4.1. Strategy **

In the remainder, the strategy is due to Margulis. Show that is Kazhdan (Shalom 2000). Show that 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 such that every point is moved at least 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 is a proper space and acts isometrically on without fixed point either on or on , then action on is uniform.

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

** 4.2. Proof of Property (T) **

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

Proposition 8Let 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 be a cocompact lattice with dense projections. Assume and are compactly generated. Let act uniformly on a Hilbert space . Then the linear part of the action has a sub-representation which extends continuously to via a projection onto one factor.

Proposition 10Let be as in Theorem 4. Then every homomorphism extends to a continuous homomorphism .

Let be a normal subgroup of . Assume is not Kazhdan. By Proposition 8, get a uniform affine action of on a Hilbert space whose linear part is irreducible. Proposition 9 implies that the linear part extends to . By assumption, , hence is compact. Since was irreducible, so is , so is finite dimensional. By Theorem 5, is a finite group, hence is finite. A finite index subgroup of acts by translation on , whence a non-trivial character. By Proposition 10, has a character, contradiction.

** 4.3. Proof of Proposition 9 **

Here is the key step.

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

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

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