  We study fine structure related to finitely supported random walks on
infinite finitely generated discrete groups, largely motivated by dimension
group techniques. The unfaithful extreme harmonic functions (defined only on
proper space-time cones), aka unfaithful pure traces, can be represented on
systems of finite support, avoiding dead ends. This motivates properties of the
random walk (WC) and of the group (SWC) which become of interest in their own
right. While all abelian groups satisfy WC, the do not satisfy SWC; however
some abelian by finite groups do satisfy the latter, and we characterize when
this occurs. In general, we determine the maximal order ideals, aka, maximal
proper space-time subcones of that generated by the group element $1$ at time
zero), and show that the corresponding quotients are stationary simple
dimension groups, and that all such can occur for the free group on two
generators. We conclude with a case study of the discrete Heisenberg group,
determining among other things, the pure traces (these are the unfaithful ones,
not arising from characters).
