Let $x$ and $y$ be two permutations of $\mathbb{Z}^2$ defined as follows. The permutation $x$ sends $(n,0)$ to $(n+1,0)$ and fixes all else while $y$ sends $(0,n)$ to $(0,n+1)$ and fixes all else. Is the group generated by $x$ and $y$ amenable?


I do know that the group does not contain a copy of the free group on two generators, so it is very likely to be amenable. I also know that if $y$ is defined, instead, by sending $(n,m)$ to $(n,m+1)$ then the group generated by $x$ and $y$ is amenable, in fact, it is a solvable extension of a locally finite group. 



Is the group generated by two almost disjoint infinite cycles amenable?
The derived subgroup of your group consists of permutations with finite support. Indeed, suppose that $w$ is a commutator word in $a$ and $b$ so the total exponent of $a$ (of $b$) is 0. Take a point $(m,n)$ where $m$ or $n$ are very large (comparing to $|w|$). Then $w(a,b)$ fixes that point. Therefore your group is an extension of a locally finite group by the Abelian group ${\mathbb Z}^2$, and is amenable. 


