A homogeneous space with no nonconstant quasiharmonic functions

Björklund asked for a countable measured group $(G,\mu)$ and a subgroup $H$ of infinite index such that the homogeneous space $G/H$ carries no nonconstant left $\mu$-harmonic or $\mu$-quasiharmonic functions, bounded or not. We give such an example. The group is the group of finitely supported permutations of $\mathbb{N}_0=\{0,1,2,\dots\}$, the subgroup is the stabiliser of $0$, and $\mu$ is any probability measure that charges every transposition $(0 n)$, $n\ge1$, and nothing else. The induced walk on $G/H\cong\mathbb{N}_0$ is a star with infinitely many leaves. The equation at a leaf $k$ forces a quasiharmonic function with defect $c$ to satisfy $u(k)=u(0)-c/\mu((0 k))$, 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.