Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word length of $g$ and $\alpha >0$).
If a subgroup $H$ of $G$ is recurrent for the random walk (meaning that the random walk will hit it "eventually"), there is a hitting measure $\eta$ on $H$ given by $\eta(h) =$ the probability that a random walk started at the identity will hit $H$ for the first time at $h$.
For example if one looks at the case where $H \lhd G$ and $G/H \simeq \mathbb{Z}$, then there is a bound simply by estimating the length of the walk before return (using Catalan numbers, I get a length of $n$ with probability asymptotic to $n^{-3/2}$). This means in that case the hitting measure $\eta$ on $H$ has finite $\alpha$-moment for $\alpha <1/2$.
However this is not optimal in general: consider the lamplighter (with lamp group $K$) $G = K \wr \mathbb{Z}$ with $H = \oplus_{\mathbb{Z}} K$. If $K$ is finite, then the measure $\eta$ has finite moment for $\alpha <1$ (since the range of the walk on $\mathbb{Z}$ suffices for an upper bound on the length of the elements).
$\mathbf{Question \; 1:}$ which moments are finite for $G = K \wr \mathbb{Z}$ (depending on $K$)?
I would guess the $1/2$ bound to be correct when $K$ has linear drift (e.g. free group on two generators), but for other groups (e.g. $K = \mathbb{Z}$) higher moments could be finite (I'm guessing $\alpha<1$). It seems that most estimates should be already available, but I could not find them and would appreciate being pointed in the right direction.
Some simpler case (like $\mathbb{Z} = H < G = \mathbb{Z}^2$) then the hitting measure could have finite $\alpha$-moments for $\alpha <2$. On the other hand, as soon as one looks at $G = K \wr \mathbb{Z}^2$ then the $\alpha$-moments are no longer finite (even when $K$ is finite).
But in general $H$ is not necessarily a normal subgroup. Then estimates on the hitting measure can be obtained by looking at the random walk on the Schreier Graph of $G/H$. There are Schreier graphs so that the random walk still returns more quickly than on $\mathbb{Z}^2$ but less than on $\mathbb{Z}$. Hence my second question:
$\mathbf{Question \; 2:}$ are there relatively natural examples of pairs $G$ and $H$ where the hitting measure is finite, but the Schreier graph of $G/H$ is not [quasi-isometric to a] line?
The reason for the "relatively natural" is that for (almost) any regular graph, one can cook up a pair $G$ and $H$ which admits this graph as a Schreier graph. And there are regular graphs where the behaviour of the return to the root is between that $\mathbb{Z}$ and $\mathbb{Z}^2$. So "relatively natural" here means, not explicitly designed for the purpose of the question.
