We discuss a flexible construction of groups where the speed (rate of escape) of simple random walk can follow any sufficiently nice function between diffusive and linear. When the speed of the \mu-random walk is sub-linear, all bounded \mu-harmonic functions are constant. We investigate the minimal growth of non-constant harmonic functions on these groups and show it is tightly related to the speed of the random walk. Based on joint works with Jeremie Brieussel, Gidi Amir and Gady Kozma.