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Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-fuzz$^{***}$ of $\Gamma$ such that the Poisson boundary of $\Gamma'$ is trivial but the growth of balls in $\Gamma'$ is still fast (say faster than any polynomial, and uniformly so at each point)$^{****}$.

$^{*}$though I lack precise arguments in this direction, I believe the answer to this question is "no" if the group is not amenable.

$^{**}$there are results (Kaimanovich) which show (amenable) wreath products such as $\mathbb{Z} \wr \mathbb{Z}$ or $\mathbb{Z}_2 \wr \mathbb{Z}^n$ (with $n \geq 3$) have non-trivial Poisson boundary. Outside wreath products, are there any other amenable groups where this phenomenon is known to occur (for any finitely supported measure)?

$^{***}$a $k$-fuzz of $\Gamma$ is obtained by from $\Gamma$ by adding edges between all points at distance $\leq k$.

$^{****}$ There's a result of B.~Seward which shows amenable groups possess a spanning line (so trivial Poisson boundary). Adding edges to this line seems to be a good idea (say to get a growth of type $e^\sqrt{n}$, but then the resulting graph is no longer a Cayley graph (...and subexponential growth does not guarantee vanishing of the Poisson boundary?)

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  • $\begingroup$ Actually, because of the Guivarc'h inequality $h\leq lv$, subexponential growth (i.e. $v=0$) implies that $h=0$ which in turns implies that the Poisson boundary is trivial. $\endgroup$
    – M. Dus
    Commented Apr 11, 2022 at 12:29
  • $\begingroup$ @M.Dus oh, is this true for graphs which are not vertex-transitive? $\endgroup$
    – ARG
    Commented Apr 12, 2022 at 6:56
  • $\begingroup$ I have to think about it, but I think so. The sketch of proof is very simple. The random walk at time $n$ asymptotically lives in a set of cardinality $\mathrm{e}^{hn}$. Its distance to the origin is asymptotically given by $nl$ so it is contained in a ball of radius $nl$ which has cardinality $\mathrm{e}^{vln}$. Hence, $h\leq lv$. $\endgroup$
    – M. Dus
    Commented Apr 12, 2022 at 8:58
  • $\begingroup$ @M.Dus yes, you're right this bound is fairly generic. Still I'm not sure if it is easy to add edges (starting from the spanning line), in order to insure both a uniform superpolynomial lower bound on the volume and a uniform subexponential upper bound. If so then that would answer my question... $\endgroup$
    – ARG
    Commented Apr 12, 2022 at 19:20

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I don't think much is currently known. Concerning $(\ast\ast)$, there are also finitely presented groups with this property (although they are essentially obtained from wreath products) - see Erschler, Liouville property for groups and manifolds, Invent. Math. 155 (2004). As for $(\ast\ast\ast)$, it is a particular case of the (widely open) question about stability of the Liouville property for finitely generated groups. Namely, whether all simple random walks on a fixed group are Liouville or not simultaneously (i.e., that the Liouville property does not depend on the generating set). In a stronger form this question can be asked about all random walks with a finitely supported non-degenerate symmetric step distribution. It is known that the Liouville property is not stable with respect to rough isometries for simple random walks on graphs or for the Brownian motion on manifolds (which is due to Terry Lyons, 1987), however all known examples are very far from being group invariant.

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  • $\begingroup$ Thanks! Actually, at the beginning of the article she mentions "Since the groups in question are amenable, they do admit measures such that the corresponding random walk has trivial boundary." What is the reference for this property? $\endgroup$
    – ARG
    Commented Mar 1, 2013 at 15:02
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    $\begingroup$ This is a theorem of Kaimanovich-Vershik-Rosenblatt: a group is amenable iff it carries a Liouville random walk. However, the step distribution of this random walk need not be finitely supported (for instance, for higher dimensional lamplighter groups). She proves in that paper that actually for these groups all measures with finite entropy have non-trivial Poisson boundary. $\endgroup$
    – R W
    Commented Mar 1, 2013 at 16:10

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