Hyperbolic groups have flat-rank at most 1

Abstract

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flatrank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.

Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group

“…Cayley-Abels graphs are unique only up to quasi-isometry however and, although they can be used to derive bounds on integer invariants of t.d.l.c. groups, see ([30] Proposition 4.6) and [42,43], they do not provide an effective method for performing precise calculations of invariants such as the scale unless the graph structure is understood in as much detail as it is for buildings. The limitation of the Cayley-Abels geometric representation therefore is that, unlike the cases of geometries for finite and Lie groups, it is not well understood for general t.d.l.c.…”

Simple groups of automorphisms of trees determined by their actions on finite subtrees

“…• Lie groups over fields of p-adic numbers and over fields of formal Laurent series over some finite residue field, where the scale is a power of the characteristic of the residue field and the flat-rank equals the usual algebraic rank; • completions of Kac-Moody groups over finite fields [9] [8], where again the scale is a power of the characteristic of the residue field and the flat-rank equals the algebraic rank [5]; • automorphism groups of buildings with negative curvature [15], where the flat-rank is at most 1 [4]; • groups of almost automorphisms of trees [16], where the flat-rank is infinite and the scale depends on valencies of the tree [23]; and…”

Quasi-actions and generalised Cayley–Abels graphs of locally compact groups

“…Similar concepts have been fruitfully employed in the study of geometric group theory in the case of discrete groups (see for example [5,9]), and one would expect that the corresponding concepts will be useful in the study of locally compact groups. One can study coarse geometry of compactly generated locally compact groups using the word-length metric with respect to a compact generating set, as done for example in [4,2]. The rough Cayley graph ignores the local information but retains the large-scale information thereby giving an alternative, but equivalent, approach to coarse geometry of locally compact groups.…”

“…In [8] Krön and Möller defined the rough Cayley graph of a topological group G to be a connected graph X such that G acts transitively on the set of vertices of X and the stabilisers of the vertices are compact open subgroups of G. We shall follow the terminology of [2] where these graphs were renamed as relative Cayley graphs. Every totally disconnected, compactly generated, locally compact group G has a locally finite relative Cayley graph: the vertex set of such a graph can be realised as the homogeneous space G/U with respect to a compact open subgroup U.…”

“…Every totally disconnected, compactly generated, locally compact group G has a locally finite relative Cayley graph: the vertex set of such a graph can be realised as the homogeneous space G/U with respect to a compact open subgroup U. In [2] relative Cayley graphs were used to show that hyperbolic totally disconnected groups have flat rank at most 1. In the case of totally disconnected, compactly generated, locally compact groups, the relative Cayley graph fits into our notion of rough Cayley graph.…”