To every topologically transitive Cantor dynamical system $(X, \varphi)$ we associate a group $T(\varphi)$ acting faithfully by homeomorphism on the real line. It is defined as the group of homeomorphisms of the suspension flow of $(X, \varphi)$ which preserve every leaf and acts by dyadic piecewise linear homeomorphisms in the flow direction. We show that if $(X, \varphi)$ is minimal, the group $T(\varphi)$ is simple, and if $(X, \varphi)$ is a subshift the group $T(\varphi)$ is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system $(X, \varphi)$ is minimal, every action of the group $T(\varphi)$ on the circle has a fixed point, providing examples of so called "orderable monsters". We additionally have the following: for every subshift $(X, \varphi)$ the group $T(\varphi)$ does not have non-trivial subgroups with Kazhdan's property (T); for every 查看全文>>