We study contraction groups for automorphisms of totally disconnected locally compact groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W , we derive the inequalities: alg-rk(W ) ≤ flat-rk(G) ≤ rk(|W | 0 ). Here, alg-rk(W ) is the maximal Z-rank of abelian subgroups of W , and rk(|W | 0 ) is the maximal dimension of isometrically embedded flats in the CAT(0)-realization |W | 0 . We can prove these inequalities under weaker assumptions. We also show that for any integer n ≥ 1 there is a topologically simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.

We describe the asymptotic behavior of automorphisms of totally disconnected locally compact groups in terms of a set of 'directions' which comes equipped with a natural pseudo-metric. The structure at infinity obtained by completing the induced metric quotient space of the set of directions recovers familiar objects such as: the set of ends of the tree for the group of inner automorphisms of the group of isometries of a regular locally finite tree; and the spherical Bruhat-Tits building for the group of inner automorphisms of the set of rational points of a semisimple group over a local field.

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.

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