Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$ Let $\phi:H\rightarrow G$ be a homomorphism. We define a map $\psi:G\rightarrow G$ as follows. For $g\in H{a_i}$, let $\psi(g)=\phi(ga_i^{-1})$. Then for each $g\in G$, we can define a sequence $s(g)=(i_0i_1\dots i_k\dots)$ with $i_k\in\left\{1,2,\ldots,n\right\}$ such that for $k=0,1,\dots$ we have $\psi^k(g)\in H{a_{i_k}}$.
Let $\mathcal{S}=\left\{s(g)|g\in G\right\}$ be the set of all sequences which can be induced by some $g\in G$.
Question: Does $\mathcal{S}$ only contain preperiodic sequence? That is to say, for each $(i_0i_1\dots i_k)\in\mathcal{S}$,there exist $N$ and $T$ such that $i_{k+T}=i_k$ for all $k\geq N$.
It seems that the answer is positive if $G$ is a Fuchsian group, but I am not sure.
