Factor of i.i.d. processes and Cayley diagrams

Riley Thornton

Published 2020-11-30Version 1

We investigate when a $\Gamma$-f.i.i.d. solution (or approximate solution) to a local labelling problem on a Cayley graph $G=\operatorname{Cay}(\Gamma, E)$ lifts to an $\operatorname{Aut}(G)$-f.i.i.d. solution (or approximate solution). A Cayley diagram is an edge labelling of $G$ by elements of the generating set $E$ so that the cycles correspond exactly to paths labelled by relations in $\Gamma$. We show that if $G$ admits an $\operatorname{Aut}(G)$-f.i.i.d. (approximate) Cayley diagram, then any $\Gamma$-f.i.i.d. (approximate) solution to a local combinatorial problem lifts to $\operatorname{Aut}(G)$-f.i.i.d. (approximate) solution. We also establish a number of results on which graphs admit such a Cayley diagram. We show that Cayley graphs of amenable groups and free groups always admit $\operatorname{Aut}(G)$-f.i.i.d. approximate Cayley diagrams, and that torsion-free nilpotent groups never admit nontrivial $\operatorname{Aut}(G)$-f.i.i.d. Cayley diagrams. Finally, we construct a Cayley graph with a $\Gamma$-f.i.i.d. 3-colorings that does not lift to even an approximate $\operatorname{Aut}(G)$-f.i.i.d. 3-coloring. Our construction also answers a question of Weilacher.