<p><span style="font-size: x-small;"><span style="font-size: 10pt;">The integers (while
wonderful in many others respects) do not make for fascinating Geometric
Group Theory. They are, however, essentially the only infinite finitely
generated group which is both hyperbolic and amenable. In the class of
locally compact topological
groups, the intersection of these two notions is richer, and the major
aim of this talk will be to give the structure of a classification of
such groups due to Caprace-de Cornulier-Monod-Tessera, beginning with
Milnor's proof that any connected Lie group admitting
a left-invariant negatively curved Riemannian metric is necessarily
soluble.</span></span></p>