## Notes of David Fischer’s Cambridge lecture 11-05-2017

Zimmer’s conjecture

Joint work with Aaron Brown and Sebastian Hurtado.

1. Actions of lattices on manifolds

${G}$ is a higher rank simple Lie group, ${\Gamma}$ a lattice in ${G}$. From Margulis’ superrigidity theorem, we know the classification of unitary representations of ${\Gamma}$. In particular, there are no (infinite image) representations in dimension less than the rank.

In analogy, Zimmer 1983 asked wether one could classify smooth actions of lattices on compact manifolds. He conjectured that no (infinite image) actions exist if ${dim(M), no volume preserving actions if ${dim(M)=r}$.

Examples.

• ${Sl(n,{\mathbb Z})}$ acts on ${{\mathbb R} P^{n-1}}$.
• ${Sl(n,{\mathbb Z})}$ acts on ${n}$-torus in a volume preserving manner.

Theorem 1 Zimmer’s conjecture is true for cocompact lattices, with the sharp dimension bound for ${Sl}$ and ${Sp}$ (a not quite sharp bound yet for other root systems).

Conjecture was known for homeos of the circle (Witte-Morris for certain lattices, Ghys, Burger-Monod in general), for surfaces (Polterovich, Franks-Handel, non-uniform lattices).

2. Towards the proof

2.1. Why Zimmer believed in it

Zimmers cocycle superrigidity theorem implies the following. Let ${\Gamma}$ act smoothly on ${M}$. Then for any invariant measure, ${\rho(\Gamma)}$ preserves a measurable Riemannian metric ${g}$. If ${g}$ were bounded, the conjecture would follow easily. In general, ${g}$ need not define a distance on ${M}$.

2.2. A priori estimate

Let ${\Gamma}$ act smoothly on ${M}$. Let ${S}$ be a fnite genrating set. Fix a reference smooth Riemannian metric ${g_0}$. Then chain rule implies that ${|D\rho(\gamma)|_{g_0}}$ grows at most exponentially.

Preserving ${g}$ implies that the first Lyapunov exponent

$\displaystyle \begin{array}{rcl} \lim_{|\gamma|\rightarrow\infty}\frac{\log|D\rho(\gamma)_x|_{g_0}}{|\gamma|}=0 \end{array}$

${\mu}$-almost everywhere. Can we exploit this?

Definition 2 Say that a smooth action ${\rho}$ has subexponential growth of derivatives if ${\forall \epsilon>0}$; ${\exists C}$ such that

$\displaystyle \begin{array}{rcl} |D\rho(\gamma)_x|_{g_0}\leq C\,e^{\epsilon|\gamma|} \end{array}$

uniformly in ${x}$.

This does not follow formally from vanishing of the Lyapunov exponent. Nevertheless, the following is encouraging:

Proposition 3 A ${{\mathbb Z}}$ action ${\rho}$ has subexponential growth of derivatives iff it has 0 Lyapunov exponent for all invariant measures.

Indeed, apply assumption to weak limits of atomic measures on long segments of orbits. Apply Kingman’s subadditive theorem and the chain rule.

Since lattices are nonamenable, no chance for such a trick to work directly.

2.3. Recipe

1. Prove subexponential growth of derivatives using
1. hyperbolic dynamics
2. homogeneous dynamics (Ratner, Shah).
3. nonlinear measure rigidity (Brown-Rodriguez-Hertz-Wang).
2. Find an invariant smooth metric using
1. Strong property (T) (Lafforgue, de la Salle-de Laat),
2. Sobolev spaces, estimates on composition (Fischer-Margulis).

2.4. Ratner? Where are the unipotents?

Use suspension, a ${G}$ action, and unipotents of ${G}$. Pick a ${K}$-invariant metric. Subexponential growth of derivatives for ${\Gamma}$ is equivalent to subexponential growth of fiber derivatives for ${G}$. This is where we need ${\Gamma}$ to be cocompact. The upshot is an ${A}$-invariant ergodic measure ${\mu}$ and an element of ${A}$ which has positive Lyapunov exponent.

Next step is to promote ${\mu}$ to a ${G}$-invariant measure ${\mu'}$, preserving positivity of Lyapunov exponent. This contradicts cocycle superrigidity.

For this,

1. Use Ratner and Shah to average ${\mu}$ over unipotent subgroups so the result ${\mu'}$ projects to Haar measure. This is the harder part of the proof.
2. Non-linear measure rigidity implies that ${\mu'}$ is ${G}$-invariant. They do the case where ${\mu'}$ is initially ${P}$-invariant, we merely need to add one algebraic lemma.

I recommend Ledrappier-Young’s approach to non-linear measure rigidity.

2.5. Strong property (T)

Lafforgue defines ${\epsilon}$-subexponential norm growth for actions on Hilbert spaces as follows: ${\exists C}$ such that

$\displaystyle \begin{array}{rcl} |\rho(\gamma)|\leq C\,e^{\epsilon|\gamma|}. \end{array}$

Fix a measure on ${\Gamma}$. Lafforgue (in fact, de la Salle) says that ${\Gamma}$ has strong property (T) if ${\exists \epsilon_0}$ such that for all smaller ${\epsilon}$ and all actions ${\pi}$ with ${\epsilon}$-subexponential norm growth, ${\pi(\mu^{\star n})}$ converges exponentially fast to projection on invariant vectors.

We apply this to the action of ${\Gamma}$ on Sobolev spaces of metrics,${W^{2,k}(M,S^{2}T^*M)}$. Using estimates on compositions, subexponential growth of derivatives implies subexponential norm growth on these Hilbert spaces. Starting from ${g_0}$, ${\rho(\mu^{\star n})g_0}$ converge to a ${\Gamma}$-invariant metric. Why is it definite? Because is not far from ${g_0}$.

3. Questions

Valette: do you need the orthogonal projection? No, I take whatever projection Lafforgue gives me.