I've been going through a paper by Warshall where he shows the presence of dead-ends of unbounded depth$^1$ (for any generating set) in the (discrete) Heisenberg group. This turns out to be the case for the lamplighter group [Cleary and Taback] (and perhaps for many amenable non-virtually-abelian groups?), and the lamplighter has also unbounded retreat depth$^2$ [Erschler]. The question is if one "chops off" the dead-ends, how fast does the group still grows?

More precisely, given an (finitely generated) amenable non-virtually-abelian group $G$, construct its Cayley graph for a (finite) generating set $S$. Take a function $f:G \to [0,\infty[$ with only one local minimum at $e_G$. Keep/orient the edges so that $f$ is strictly increasing by following the oriented edges. Take away all vertices $g$ so that there is no infinite oriented path starting at $e_G$ and passing through $g$.

$\textbf{Question 1}$: If $f(g) = |g|_S$, may it happen that the graph grows more slowly (i.e. little o) than the original Cayley graph?

$\textbf{Question 2}$: Does there exists an $f$ so that no (or few) vertices are removed in the process?

$^1$ if I'm quoting correctly, given a generating set $S$ of an infinite group $G$, the depth of an element $g \in G$ is $\inf \lbrace d_S(g,g') \mid g' \in B_{|g|_S} ^c \rbrace$, i.e. the distance to the closest element of strictly bigger word length.

$^2$ the retreat depth of $g \in G$ (w.r.to $S$) is the smallest $d$ such that $g$ belongs to an infinite component in the Cayley graph minus its ball of radius $B_{|g|_S-d}$. Actually, what are the (amenable non-virtually-abelian) groups with bounded retreat depth?