On the quasi-isometric classification of locally compact groups
  This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous negatively curved manifolds. A main conjecture provides a general description; an extended discussion reduces this conjecture to more specific statements.   In the course of the paper, we provide statements of quasi-isometric rigidity for general symmetric spaces of noncompact type and also discuss accessibility issues in the realm of compactly generated locally compact groups. 
Yves Cornulier
math.GR math.GT math.MG