Random walks on groups from a dimension group perspective

Abstract: Let G be a finitely generated infinite discrete group, and let S, containing 1, be a finite subset of G that generates it as a semigroup (that is, $U_{n=0}^{\infty}S^n = G$). Let P be an element of the group algebra AG (where A is either the integers or the reals), whose support is S, and all of whose nonzero coefficients are positive. Then left multiplication by P is a positive homomorphism $AG \to AG$, and iterating it leads to an unnormalized random walk on G. We can associate in the obvious way the structure of a dimension group (a direct limit of simplicially ordered torsion-free abelian groups/finite-dimensional vector spaces).

We are interested in space-time cones associated to this construction, and the harmonic functions thereon (generalizing from the case of abelian groups, a method of proving even- tual positivity for repeated multiplication by P), that reflect properties of the random walk. A natural cone arises by setting $L_n$ to be the subset of G that can be reached by n iterates of S starting at 1, i.e., $L_n = S^n$, and this has the advantage that at each stage, we are dealing with finite-dimensional vector spaces. However, this is still quite complicated and massively redundant; so we define Ln to be Sn with all points reached in fewer than n steps deleted. This is better from the dimension group point of view, but there is now the possibility of dead-ends, that is, g in $L_n$ with $g · S\subset   S^n$ (so no gs—with s in S—belongs to $L_{n+1} · S$), and these occur almost ubiquitously.

We first describe how we can refine $L_n$ to avoid dead-ends without loss of information, and then study properties of the random walk that are naturally suggested by behaviour of these new (almost-) partitions of G. Then we apply them to torsion-free abelian by finite groups, and show that some are much better behaved than others, by considering the induced integral action. Then we discuss other groups, and in some cases, determine the pure (= extremal = indecomposable = ergodic) unfaithful finite harmonic functions on them, in particular, for the simplest discrete Heisenberg group and the lamplighter group. Finally, we show that the quotients by the maximal order ideals of the resulting dimension groups are always ordinary stationery dimension groups (and if A is the integers, every such can be obtained for some free group and choice of P), so in particular, have unique trace. In the case of the lamplighter group, this exhausts the unfaithful pure harmonic functions, but in the case of the Heisenberg group, don’t even amount to a hill of traces.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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