We introduce the discrete affine group of a regular tree as finitely generated subgroup group. describe Poisson boundary random walks on it space configurations. compute isoperimetric profile and Hilbert compression exponent also discuss metric relationship with some lamplighter groups graphs.

Consider a locally compact group G=Q⋉V such that V is abelian and the action of Q on dual Vˆ has free orbit full measure. We show G can be quantized in three equivalent ways: by reflecting across Galois object defined canonical irreducible representation L2(V); twisting coproduct von Neumann algebra 2-cocycle obtained from G-equivariant Kohn–Nirenberg quantization V×Vˆ; considering bicrossed pr...