Just started learning the StoneCech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature devoted to $\beta\mathbb{N}$ than to $\beta\mathbb{Z}$. I wonder why is that? After all, algebraically $\mathbb{N}$ is a semigroup while $\mathbb{Z}$ is a group, and as discrete topological spaces they are homeomorphic. From your experience, how far $\beta\mathbb{N}$ and $\beta\mathbb{Z}$ are different in behaviour? Also, is $\beta\mathbb{N}$ ( or ($\beta\mathbb{N}\setminus\mathbb{N}$) easier to deal with?

$\beta\mathbb Z$ consists of just two copies of $\beta\mathbb N$, one at "positive infinity" and one at "negative infinity". It's generally easier to think about just one copy rather than both, so people tend to write more about $\beta\mathbb N$. The advantage you mentioned for $\mathbb Z$ over $\mathbb N$, namely that the former is a group while the latter is only a semigroup, doesn't carry over to the StoneCech compactifications, both of which are (under the natural extensions of the addition operation) only semigroups. 


Since $\mathbb N$ and $\mathbb Z$ are homeomorphic, so are $\beta\mathbb N$ and $\beta\mathbb Z$, though of course the semigroup structure will be different. 

