11 problems

Let $G$ be a higher rank group, let $\Gamma$ be a Borel Anosov subgroup of $G$, and let $\psi_\Gamma$ and $\rho$ denote the growth indicator function of $\Gamma$ and the relevant c…

Higher-rank homology growth conjecture. For every prime $p$ and every $n\in\\{0,\dots,r-1\\}$,

Let $G$ be a connected semisimple Lie group with higher rank, finite center, and no compact factors, and let $\Gamma$ be an irreducible lattice in $G$. Serre's congruence subgroup…

Let $G_0$ be a real linear simple Lie group of real rank at least $2$, let $K$ be a maximal compact subgroup of $G_0$, and let $\Gamma$ be a discrete subgroup of $G_0\times G_0$ ac…

Let $\Gamma$ be the arithmetic group specified by the local-to-global conjecture above. A group $G$ is flexibly stable if every sufficiently accurate approximate homomorphism from…

Let $\Gamma$ be a discrete subgroup in the setting of higher-rank locally symmetric spaces, and call its joint $L^2$-eigenvalues complementary eigenvalues when they lie in the comp…

Let $G$subset be a higher rank semisimple real algebraic group and let $\Gamma$ be an Anosov subgroup of $G$. Denote by $\psi_\Gamma$ its growth indicator and by $\rho$ the half-su…

Let $\Gamma$ be a noncocompact irreducible arithmetic group, and let $\operatorname{rank}_{\mathbb{R}}\Gamma$ denote its real rank. A group is boundedly generated by unipotent subg…

Let $\Gamma$ be a noncocompact arithmetic subgroup of either $\operatorname{SL}(3,\mathbb{R})$ or $\operatorname{SL}(3,\mathbb{C})$. A faithful action on $\mathbb{R}$ means an acti…

Let $\Gamma$ be a non-cocompact lattice in either $\mathop{\mathrm{SL}}\nolimits(3,\mathbb{R})$ or $\mathop{\mathrm{SL}}\nolimits(3,\mathbb{C})$. Non-cocompact rank-three conjectur…

Sarnak's conjecture. Every such $\Gamma$ is essentially cuspidal: equality holds in the displayed Weyl-law bound. If true, this would show that the cuspidal spectrum has full Weyl-…