A new lattice invariant for lattices in totally disconnected locally compact groups (Q2218725)

A property is called a \emph{lattice invariant} if whenever \(\Gamma\) and \(\Gamma'\) are lattices in a locally compact \(G\), then \(\Gamma\) has the property if and only if \(\Gamma'\) does. Every measure equivalence invariant is also a lattice invariant, including Kazhdan's Property (T) and amenability, but lattice invariants that are not measure equivalence invariants are rare. The authors introduce such an invariant for \(\sigma\)-compact totally disconnected locally compact groups, called \emph{bounded conjugacy rank}. It takes values in \(\mathbb{N}_{0}\cup\{\infty\}\). For example, the rank of the automorphism group of a regular tree is zero, and groups of positive or infinite rank are exhibited as well. Finally, the new invariant is employed to construct more totally disconnected locally compact groups without lattices, and compared to the non-lattice invariant but related property of \emph{inner amenability}.