TITLE: Arithmetic progressions in finitely generated groups
QUESTION [7 upvotes]: Suppose $\Gamma$ is a finitely generated countable discrete torsion free group with a generating set $S$. Let $l$ be the word length function given by $S$. 
Let $F_n=\{s\in\Gamma| l(s)\leq n\}$.
Assume that $\Lambda$ is a subset of $\Gamma$ such that $$\limsup_{n\to\infty} \frac{|\Lambda\cap F_n|}{|F_n|}>0.$$
Question:  For every positive integer $k$, do there exist $b$ and $a$ in $\Gamma$ such that $\{b^{j}a\}_{j=0}^{k-1}\subseteq\Lambda$?
When $\{b^{j}a\}_{j=0}^{k-1}$has $k$ distinct elements, we call it a left arithmetic progression of length $k$ in $\Gamma$.
Remark: One would avoid the trivial case that $b=e_\Gamma$. In fact, let $\Gamma=\mathbb{Z}$ with $S=\{1\}$, the answer to the above question is affirmative by Szemeredi's theorem, which says that a subset of $\mathbb{Z}$ with positive upper density contains arbitrarily long arithmetic progressions.

REPLY [2 votes]: Your definition of "arithmetic progression" in a group is wrong. It should be as follows. Let  $w(x,a_1,...,a_n)$ be a word. Then an arithmetic progression is the sequence $w(a_1,a_1,...,a_n), ..., w(a_n,a_1,...,a_n)$ for some a_1,...,a_n in the group (see https://en.wikipedia.org/wiki/Hales%E2%80%93Jewett_theorem). In this formulation the result is true as proved by Furstenberg and Katznelson (it was a \$100 problem from a list by Ron Graham, as far as I remember, http://www.math.ucsd.edu/~ronspubs/08_06_old_and_new.pdf).