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.
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 6 Let 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.
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 8 Let 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 10 Let 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.