On solvable minimally transitive permutation groups

We investigate properties of finite transitive permutation groups $(G, \Omega)$ in which all proper subgroups of $G$ act intransitively on $\Omega.$ In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups.Comment: 8 pages, no figures, journal pape

Minimal generation of transitive permutation groups

It is proved in [21] that there is a constant $c$ such that each transitive permutation group of degree $d\ge 2$ can be generated by $\lfloor cd/\sqrt{\log{d}}\rfloor$ elements. In this paper, we explicitly estimate $c$

Groups acting on trees with almost prescribed local action

We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually one) not containing lattices. We also obtain examples of simple groups with simple lattices, and we prove the existence of (infinitely many) finitely generated simple groups of asymptotic dimension one. We also prove various properties of these groups, including the existence of a proper action on a CAT(0) cube complex.Comment: v2: 35 pages; argument slightly modified in 4.2.2; final versio

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.