9 problems

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, let $\{p_1,\ldots,p_n\}$ be a nonempty set of prime numbers, and write…

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, and let $G_{\mathord{\mathbb{Z}}}$ denote its integral points. A subgroup is left orderable i…

Local indicability conjecture. Every left ordered $\operatorname{NF}$-group is locally indicable.

Let $g\geq 2$ and let $b_1,\ldots,b_n$ be curves isotopic to the boundary components of the surface $\Sigma_g^n$. For positive exponents $k_1,\ldots,k_n$, consider the product … He…

Let $M$ be a rational homology $3$-sphere, meaning that $H_1(M;\mathbf{Q})=0$. Its Heegaard Floer homology has rank at least $\left|H_1(M;\mathbf{Z})\right|$, and $M$ is an L-space…

Boyer–Gordon–Watson L-space conjecture. $M$ is not an L-space if and only if $\pi_1(M)$ is left-orderable.

Let $\Gamma$ be a group. A total order $\prec$ on $\Gamma$ is left-invariant if $a\prec b$ implies $ca\prec cb$ for all $a,b,c\in\Gamma$. The algebraic version of the conjecture. I…

Let $G$ be a connected semisimple Lie group with finite center and real rank at least $2$, and let $\Gamma$ be an irreducible lattice in $G$. High-rank lattice action conjecture. T…

Let $\Gamma$ be a lattice in $\mathop{\mathrm{SL}}\nolimits(n,\mathbb{R})$, with $n\geq 3$. A group is left orderable if it admits a left-invariant total order. Left-orderability c…