5 problems

Finite-orbit conjecture. For any Cayley graph of $G$, the horofunction boundary contains a finite orbit for the canonical group action.

Finite-dimensionality characterization conjecture. The group $G$ has polynomial growth if and only if $$ is finite dimensional.

Harmonic-function characterization. The following conditions are equivalent:

Countability conjecture. The set of horofunctions on $G$ is countable.

Let $\mathcal G$ be a boundary groupoid with polynomial volume growth. The smooth extension conjecture. Every such boundary groupoid satisfies the smooth extension property: for an…