Dynamical Tits alternative for groups of almost automorphisms of trees

We prove a dynamical variant of the Tits alternative for the group of almost automorphisms of a locally finite tree 𝒯: a group of almost automorphisms of 𝒯 either contains a nonabelian free group playing ping-pong on the boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="0em">∂</m:mo> <m:mi mathvariant="script">T</m:mi> </m:mrow> </m:math> , or the action of the group on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="0em">∂</m:mo> <m:mi mathvariant="script">T</m:mi> </m:mrow> </m:math> preserves a probability measure. This generalizes to all groups of tree almost automorphisms a result of S. Hurtado and E. Militon for Thompson’s group 𝑉, with a hopefully simpler proof.