Line graphs and 22-geodesic transitivity

For a graph $\Gamma$, a positive integer $s$ and a subgroup G\leq \Aut(\Gamma), we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive

High transitivity for more groups acting on trees

We establish a new sharp sufficient condition for groups acting on trees to be highly transitive. This give new examples of highly transitive groups, including icc non-solvable Baumslag-Solitar groups, thus answering a question of Hull and Osin.Comment: Version 2 : correction of a mistake in an example from section 8.3 and general improvement of section 8.