A homogeneous space with no nonconstant quasiharmonic functions

Björklund asked for a countable measured group (G,μ) and a subgroup H of infinite index such that the homogeneous space $G/H$ carries no nonconstant left μ-harmonic or μ-quasiharmonic functions, bounded or not. We give such an example. The group is the group of finitely supported permutations of N₀=\0,1,2,\, the subgroup is the stabiliser of $0$, and μ is any probability measure that charges every transposition $(0 n)$, n≥1, and nothing else. The induced walk on G/H₀ is a star with infinitely many leaves. The equation at a leaf k forces a quasiharmonic function with defect c to satisfy u(k)=u(0)-c/μ((0 k)), and the equation at the centre then forces $c=0$ and u constant. The argument uses only the convergence of the single series that defines the equation at the centre.