A new lattice invariant for lattices in totally disconnected locally compact groups

We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.