7 problems

Let $E$ be a Euclidean space of dimension $d\geq 3$, and let $\nu$ be a totally strongly irreducible probability distribution supported on a discrete subgroup of $\mathrm{SL}(E)$.…

Let $G=\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}^d$ with $d\geq 1$, and let $\mu$ be a non-degenerate finitely supported measure on $G$ whose projection to $\mathbb{Z}^d$ induces a trans…

Converse conjecture. The polynomial $p$ has the spaced polynomial property.

Let $G$ be a linear basic block of dimension $2$, and consider random walks on $G$ in the setting discussed around the cited examples. The two-dimensional basic-block conjecture. T…

Let $G$ be a linear group, and let $F_G(n)$ denote its Følner function. Consider simple random walks on $G$. The Følner-growth conjecture. Simple random walks on linear groups are…

Let $G$ be a linear group in the setting of Corollary C, and consider simple or centered finite second-moment random walks on $G$. The necessity conjecture. The assumption of Corol…

Let $(G,\rho)$ be a transient stationary random graph that is almost surely planar. A stationary embedding is an embedding of $(G,\rho)$ as a stationary subgraph of the Poincaré di…