
## 2. Higher rank lattice actions

1. ### Actions on $n$ manifolds by lattices in $\textrm{SL}_{n}(\mathbb{R})$

#### Problem 2.05.

[D. Fisher] Let $\Gamma$ be a lattice in $\textrm{SL}_{n}(\mathbb{R})$. Let $\alpha:\Gamma\to \textrm{Diff}(M)$ be a smooth action on $n$ manifold $M$ by $\Gamma$.

Assume that there is an isolated periodic point with derivative is unbounded.
1. Is $\alpha$ linearizable nearby the fixed point?
2. Is there an $\alpha$-invariant open neighborhood $U$ of the fixed point in $M$ such that the restriction of $\alpha$ on $U$ is conjugate to the linear action of $\Gamma$ on $\mathbb{R}^{n}$?
1. Remark. [D. Fisher] Local linearization for analytic actions near any (not necessarily isolated) fixed point in all dimensions is contained in the paper of Carins-Ghys ([MR1460126]). Interestingly for problem 2.15 below, this is proven by giving a local extension to a $G$ action and then locally linearizing the $G$ action.

It would be interesting to have local linearization for smooth actions in dimension $n$. It seems unlikely to hold in all dimensions as Cairn-Ghys give a non-linearizable action of $\textrm{SL}(3,\mathbb{R})$ near a fixed point in $\mathbb{R}^8$. They do not prove that the restriction of that action to a lattice is non-linearizable, but probably it is not.

It continues to seem hard to see why the fixed point being isolated makes the linearization hold globally on a neighborhood, but this is true in all known examples.
• #### Problem 2.1.

[K. Melnick] Let $\Gamma$ be a lattice in $G=\textrm{SL}_{n}(\mathbb{R})$. Let $\alpha:\Gamma\to \textrm{Diff}(M)$ be a smooth action on $n$ manifold $M$ by $\Gamma$.

Assume that $\alpha$ preserves projective structure but there is no affine invariant connection. Does the action $\alpha$ extend to projective structure preserving $G$ action?

Here, a projective structure is a projective equivalence class of affine connections.
1. Remark. When $\textrm{dim}(M)=n-1$, the same question is answered by V. Pecastaing.
• #### Problem 2.15.

[D. Fisher] Let $\Gamma$ be a lattice in $\textrm{SL}_{n}(\mathbb{R})$. Let $\alpha:\Gamma\to \textrm{Diff}(M)$ be a smooth action on $n$ manifold $M$ by $\Gamma$.

Assume that there is an $\alpha$ invariant embedded $\mathbb{R}P^{n-1}$ (or $S^{n-1}$) in $M$ which $\Gamma$ acts standard way. Does $\alpha$ extend to a local $G$ action on a neighborhood of $\mathbb{R}P^{n-1}$ (or $S^{n-1}$)?
• ### Irreducible lattices in product of rank one groups

#### Problem 2.2.

[A. Brown] Let $\Gamma=\textrm{SL}_{2}(\mathbb{Z}[\sqrt{2}])$. $\Gamma$ acts on $4$-torus by automorphism using $\mathbb{Q}$-irreducible representation defined over $\mathbb{Q}$. Is it locally rigid? More generally, any Anosov $\Gamma$ actions on torus is conjugate to an affine action?
1. Remark. When $\Gamma$ acts on $\mathbb{T}^{8}$ or $\mathbb{T}^{12}$ via $\mathbb{Q}$-irreducible representation, it is locally rigid by [MR3702679] with a modification.
• Remark. [org.aimpl.user:hl63@indiana.edu] When perturbed $\Gamma$ action is volume preserving and induced $\textrm{SL}(2,\mathbb{R})\times \textrm{SL}(2,\mathbb{R})$ action is irreducible then it is smoothly conjugate with the original affine action. ([arXiv:2002.02485])
• ### Actions on manifolds by lattices in p-adic group

#### Problem 2.25.

[A. Brown] Let $\Gamma$ be a lattice in $\textrm{SL}_{n}(\mathbb{Q}_{p})$ with $n\ge 3$. Does $\Gamma$ act smoothly non-isometrically on a closed manifold?
• ### Lattices in infinite center Lie group

#### Problem 2.3.

[W. Van Limbeek] Let $\Gamma$ be a lattice in $G=\widetilde{\textrm{Sp}_{2g}(\mathbb{R})}$.
1. Does $G$ act faithfully on a compact manifold?
2. If not, does $\Gamma$ act faithfully on a compact manifold?
1. Remark. [Juan Arosemena Serrato and Bertrand Deroin] We report here on our group session on the problem posed by Wouter where we found a positive answer to its question 1: we construct a faithful analytic action of $\widetilde{\text{Sp}}_{2g} (\mathbb R)$ on a closed manifold.

We equip $\mathbb R^{2g}$ with a symplectic form, and we let $\Lambda \subset \text{Gr}_g (\mathbb R^{2g})$ be the submanifold of the $g$-dimensional Grassmannian of $\mathbb R^{2g}$ formed by Lagrangian subspaces of $\mathbb R^{2g}$. This is a smooth closed manifold invariant by the action of $\text{Sp} (2g,\mathbb R)$.

The fundamental group of $\Lambda$ is isomorphic to $\mathbb Z$, and its action on the universal cover $\tilde{\Lambda}$ of $\Lambda$ extends to a faithful action of $\widetilde{\text{Sp}} (2g,\mathbb R)$ on $\tilde{\Lambda}$, that lifts the (non faithful) one of $\text{Sp}(2g, \mathbb R)$ on $\Lambda$. This comes from the fact that the maximal compact subgroup of $\text{Sp}(2g, \mathbb Z)$ is the unitary group $U(g)$, and that $\Lambda \simeq U(g) / O(g)$. It is important to notice that the center of $\widetilde{\text{Sp}} (2g,\mathbb R)$ is isomorphic to $\mathbb Z$ and acts by Deck transformation of the covering $\tilde{\Lambda} \rightarrow \Lambda$.

Take any analytic diffeomorphism $f$ of the circle of infinite order, and define $M$ to be the quotient of $\tilde{M}:= \tilde{\Lambda} \times \mathbb S^1$ by the $\mathbb Z$ action given by

$$(*) \ \ \ \ n\cdot ( \tilde{\lambda} ,t):= (n\cdot \tilde{\lambda}, f^n (t) )$$

for every $n\in \mathbb Z$, any $\tilde{\lambda} \in \tilde{\Lambda}$ and any $t\in \mathbb S^1$; $M$ is a closed analytic manifold.

The group $\widetilde{\text{Sp}} (2g,\mathbb R)$ acts on $\tilde{M}$ by $\tilde{g} \cdot (\tilde{\lambda}, t) := (\tilde{g} \cdot \tilde{\lambda}, t)$, commuting with the $\mathbb Z$-action (*); this defines an action of $\widetilde{\text{Sp}} (2g,\mathbb R)$ on $M$.

Notice that if $t\in \mathbb S^1$ is a non periodic point of $f$, then the set $\tilde{\Lambda}\times t$ projects on $M$ injectively, and the projection is $\widetilde{\text{Sp}} (2g,\mathbb R)$-equivariant. Since the action of $\widetilde{\text{Sp}} (2g,\mathbb R)$ is faithful on $\tilde{\Lambda}\times t$, it is then faithful on $M$ too, so this gives the desired example.

We observe that $M$ has dimension $g(g+1)/2 +1$, and a good question should be to know if this is the minimal dimension of a manifold on which $\widetilde{\text{Sp}} (2g,\mathbb R)$ acts faithfully.

Another question is to decide wether lattices of $\widetilde{\text{Sp}} (2g ,\mathbb R)$ act algebraically on real/complex algebraic manifolds or not. A last question would be to replace $\text{Sp} (2g,\mathbb R)$ by other simple algebraic Lie groups with infinite fundamental group: for instance, does the universal cover of $\text{SU} (p,q)$ act faithfully on a closed manifold?
• ### Blow up procedure

#### Problem 2.35.

[A. Brown] Let $G=\mathbb{Z},\mathbb{Z}^{k} (k\ge 2)$ or a higher rank lattice $\Gamma$. Let $\alpha:G\to \textrm{Aut}(\mathbb{T}^{n})$ be a linear Anosov action. We can blow up along a periodic orbit of $A$. Can you do blow up for infinitely many periodic orbit so that the action is still smooth and there is a nonwhere vanishing density?
1. Remark. [D. Fisher] A positive answer to problem 2.15 implies a negative answer to this problem.
• ### Complex manifold

#### Problem 2.4.

[V. Pecastaing] Let $G$ be a higher rank semisimple Lie group and $\Gamma$ be a lattice in it. Does same question hold when we consider biholomorphic actions on an almost complex manifold?
• ### Bi-Lipschitz action

#### Problem 2.45.

[A. Brown] Let $\Gamma$ be a lattice in $\textrm{SL}_{n}(\mathbb{R})$ with $n\ge 3$. Let $X$ be a manifold with $\textrm{dim}(X)\le n-2$. Let $\alpha$ be a bi-lipschitz action on $X$ by $\Gamma$. Is it finite?
• ### Cocycle over projective action

#### Problem 2.5.

[Z. Wang] Let $\Gamma$ be a lattice in $\textrm{SL}_{n}(\mathbb{R})$ with $n\ge 3$. Let $$\beta:\Gamma\times \mathbb{R}P^{n-1}\to \textrm{GL}_{d}(\mathbb{R})$$ be a cocycle over a projective action on $\mathbb{R}P^{n-1}$ by $\Gamma$. Is $\beta$ measurably cohomologous to a combination of derivative cocycle and a power of Jacobian? (Here we use the quasi-invariant normalized Haar measure on $\mathbb{R}P^{n-1}$)
• #### Problem 2.55.

[B. Deroin] Is there a locally free $\textrm{SL}_{3}(\mathbb{R})$ action on a closed $9$ dimensional manifold $X$ with at least one non-closed orbit?
• ### Property (T) group action

#### Problem 2.6.

Let $\Gamma$ be a discrete group with Kazhdan’s property (T). Let $\Sigma$ be a smooth surface and $\alpha\colon \Gamma\to \textrm{Diff}^{\infty}_{vol}(\Sigma)$ be a smooth volume preserving action on $\Sigma$ by $\Gamma$. Is $\alpha(\Gamma)$ finite?

Cite this as: AimPL: Global rigidity of actions by higher-rank groups, available at http://aimpl.org/rigidhigherrank.