Elementary totally disconnected, locally compact groups of higher complexity

The article focuses on a class of second countable groups assembled from profinite and discrete by elementary operations. We focus on a rank associated with these groups that measure their complexity, the decomposition rank. A collection of groups acting on ${\mathrm{\aleph }}_0$-regular trees is defined and used for the first construction of a group with decomposition rank ${\omega }^{\omega }+1$.