Quasi-actions on trees and Property (QFA)

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre’s Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including $SL(n,\mathbb{Z})$ for $n\ge 3$. We also give a way of thinking about Property (QFA) by breaking it down into statements about particular classes of trees.