## Notes of David Fisher’s Cambridge seminar lecture 13-06-2017

Strong property (T), subexponential growth of derivatives and invariant metrics

Theorem 1 (Brown-Fisher-Hurtado) Let ${\Gamma}$ be a cocompact lattice of ${Sl(n,{\mathbb R})}$ acting smoothly on a compact manifold of dimension ${. Then action factors through a finite group.

Definition 2 Say a smooth action ${\rho}$ of ${\\Gamma}$ on a manifold has subexponential growth of derivatives if ${\forall \epsilon>0}$, ${\exists C(\epsilon)}$ such that ${\forall x}$, ${\forall\gamma\in\Gamma}$,

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

The proof of the theorem has two steps. First establish subexponential growth of derivatives. Second prove that subexponential growth of derivatives implies finiteness. Today, I focus on the second step.

Theorem 3 If ${\Gamma}$ has strong property (T) of Lafforgue and action has subexponential growth of derivatives, then action preserves a smooth Riemannian metric.

Hence we rely on

Theorem 4 (Lafforgue, de la Salle, de la Salle-de Laat) Let ${\Gamma}$ be a lattice of a higher rank simple Lie group then ${\Gamma}$ has strong property (T) of Lafforgue.

1. What is strong property (T)?

1.1. Measure characterization of Property (T)

Property (T) states that for unitary representations without invariant vectors, every unit vector is moved a definite amount by some element of a fixed generating system.

The following is an equivalent definition. Let ${\mu}$ be a finitely supported probability measure on ${\Gamma}$. Let ${\pi:\Gamma\rightarrow U(H)}$ be a unitary representation. Let ${P}$ denote the orthogonal projector onto invariant vectors. Then ${\Gamma}$ has property (T) iff for all unitary representations,

$\displaystyle \begin{array}{rcl} \|\pi(\mu)-P\|<1. \end{array}$

Indeed, merely convexity of balls is used.

1.2. Strengthening

Strong refers to generalizing from unitary to slow exponential growth. Say a inear action by bounded operators on a Hilbert space has ${\epsilon}$-subexponential norm growth if ${\exists C}$ such that ${\forall\gamma}$,

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

Definition 5 ${\Gamma}$ has strong property (T) if ${\exists\epsilon_0}$, ${\exists}$ a sequence of measures ${\mu_n}$ such that ${\forall\epsilon<\epsilon_0}$, for all representations ${\pi}$ with ${\epsilon}$-subexponential norm growth, ${\exists \lambda<1}$ and ${C>0}$ and a projection ${P}$ onto invariant vectors such that

$\displaystyle \|\pi(\mu_n)-P\|\leq C\,\lambda^n.$

This is not Lafforgue’s definition, which encompasses a larger class of Banach spaces. However, we need it only for Hilbert spaces. There are subtle points about the definition: why does it merely provide some sequence of measures? some projection? This is what de la Salle’s proof provides us. Hopefully, the theory will develop and converge to a more robust definition.

Lafforgue’s motivation was to show that a certain strategy to prove Baum-Connes conjecture for higher rank lattices could not work. Also, to produce universal expanders, i.e. collections of graphs which do not embed uniformly in large classes of Banach spaces.

1.3. How to use it

Let ${M}$ be a compact manifold. Riemannian metrics for a convex cone in the space of section of ${S^2 T^*M}$. We use the action of ${\Gamma}$ on this vectorspace. Specifically, fix a volume form ${\omega}$ and consider ${L^2(M,S^2 T^*M,\omega)}$.

Proposition 6 Subexponential growth of derivatives implies subexponential norm growth of the action on ${L^2(M,S^2 T^*M,\omega)}$.

1.4. Estimates on compositions

Next let ${W^{2,k}(M,S^2 T^*M,\omega)}$ denote the Sobolev space of metrics with ${k}$ derivatives in ${L^2}$. Then Proposition extends to this space. This follows from estimates on compositions, to be found in Fisher-Margulis. Indeed, subexponential growth of first derivatives implies subexponential growth of all derivatives. Indeed,

$\displaystyle \begin{array}{rcl} (f\circ g)''=(g'(f'\circ g))'=g''(f'\circ g)+g'^2(f''\circ g), \end{array}$

we see that only first derivatives get multiplied, second derivatives appear only linearly.

Therefore, the representation ${\pi}$ on ${W^{2,k}(M,S^2 T^*M,\omega)}$ has ${\epsilon}$-subexponential norm growth (subexponential growth of Jacobians also enters). Sobolev embedding implies that quadratic forms there are differentiable several times.

Let ${\mu_n}$ be the probability measures provided by strong property (T). Let ${g_0}$ be some Riemannian metric, and ${g_n=\pi(\mu_n)g_0}$. Then ${g_n}$ converges to a smooth invariant nonnegative quadratic form ${g_{inv}}$. Because convergence is exponential and, for every tangent vector ${v}$, ${g_n(v,v)}$ shrinks at worst subexponentially, ${g_{inv}}$ is definite, hence a metric.

2. Questions

Why not use the ${CAT(0)}$ space of ${L^2}$ metrics, on which the action is isometric? We have tried this, this works if one has an invariant volume form to start with.