2023/04/26 by Georgakopoulos, Agelos, Wendland, Alex
#05C21 #05C50 #05C63 #05E18 #31C05 #31C12 #31C20 #60J45 #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2304.13317
We prove that every recurrent graph G quasi-isometric to ℝ admits an essentially unique Lipschitz harmonic function h. If G is vertex-transitive, then the action of Aut(G) preserves ∂ h up to a sign, a fact that we exploit to prove various combinatorial results about G. As a consequence, we prove the 2-ended case of the conjecture of Grimmett & Li that the connective constant of a non-degenerate vertex-transitive graph is at least the golden mean. Moreover, answering a question of Watkins from 1990, we construct a cubic, 2-ended, vertex-transitive graph which is not a Cayley graph.