Accessibility of partially acylindrical actions

TL;DR

This paper extends bounds on group actions on trees to partially acylindrical actions with finite height exceptions, providing counterexamples to previous conjectures and establishing sharp bounds for such actions.

Contribution

It generalizes Weidmann's bounds to actions with finite height exceptions and offers a counterexample to a prior conjecture, along with precise bounds for $k$-acylindrical actions.

Findings

Extended bounds to actions with finite height exceptions

Provided a counterexample to Weidmann's conjecture

Established sharp bounds for $k$-acylindrical group actions

Abstract

In a pervious paper Weidmann shows that there a bound on the number of orbits of edges in a tree on which a finitely generated group acts $(k,C)$-acylindrically. In this paper we extend this result to actions which are $k$-acylindrical except on a family of groups with "finite height". We also give an example which gives a negative result to a conjecture of Weidmann from the same paper and produce a sharp bound for groups acting $k$--acylindrically.