Hyperbolic groups have flat-rank at most 1 (Q1760407)

The authors call a topological group \textit{hyperbolic} if it is compactly generated and its Cayley graph with respect to some (and hence any) compact generating set is Gromov-hyperbolic. The \textit{flat-rank} of a totally disconnected locally compact group \(G\) is an integer which is an invariant of \(G\) as a topological group. The main theorem is the following:\smallskip Theorem 1: The flat-rank of a totally disconnected, locally compact, hyperbolic group is at most 1. There are two further theorems: Theorem 2: Let \(A\) be a group of automorphisms of the totally disconnected, locally compact group \(G\). Suppose that \(A\) has a hyperbolic orbit in the space of compact open subgroups of \(G\). Then the flat-rank of \(A\) is at most 1. Theorem 3: Let \(G\) be a totally disconnected, locally compact group whose space of directions is discrete. Then the flat-rank of \(G\) is at most 1. If the space of directions is not empty, then the flat-rank is exactly 1.