Not residually finite groups of intermediate growth, commensurability and non-geometricity

We construct a series of not residually finite groups of intermediate growth. In particular we construct groups of intermediate growth that are not commensurable up to finite kernels with any residually finite group. Also we show that certain properties of groups, among those solvability of the word problem and the property of having torsion of unbounded order, are not weakly geometric. The argument is based in particular on studying commensurability up to finite kernels.