12 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…

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

Let $G=\langle t,x_1,\ldots,x_n\mid tw_1\cdots tw_k=1\rangle$, where $n\geq 2$ and $w_i\in\langle x_1,\ldots,x_n\rangle$ for all $i$. An isolation conjecture for positive-word one-…

Let $M$ be an irreducible $\mathbb Z$-homology $3$-sphere, and exclude the $3$-sphere $S^3$ and the Poincaré homology sphere. A group is left-orderable if it admits a left-invarian…

Let $F$ be a finitely generated free group, and let $P$ be a finitely generated subsemigroup of $F$ such that … A group of fractions is a group expressible as $P P^{-1}$. Navas's c…

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…

L-space conjecture. $Y$ is an L-space if and only if $\pi_1(Y)$ is not left-orderable.

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…