## 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 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 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 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.