In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉Rn where the semidirect product is defined by a diagonalizable matrix of determinant one with ...

Suppose we are given a homogeneous tree Tq of degree q ≥ 3, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product (Z/2) ≀ Γ, where Γ = ∗ i=1 Z/2 is the free product group of q factors Z/2. We consider a transient random walk on a Cayley graph of (Z/2) ≀ Γ, for which we want to compute lower and upper bounds for the rate of escap...