3 problems

Let $G$ be a graph, let $\Gamma$ be a closed subgroup of $\operatorname{Aut}(G)$ acting transitively on $G$, and let $\mathbf{w}$ be the induced relative weight function. Let…

FUSF heaviness and branching conjecture. Each tree in the $\operatorname{FUSF}$ is $\Gamma$-heavy and has branching number bigger than one almost surely.

Let $G$ be a connected, locally finite graph, and let $\Gamma\subseteq\operatorname{Aut}(G)$ be quasi-transitive and nonunimodular. For each $\lambda$, let $p_c(\lambda)$ denote th…