What set do generic types define? Let $G$ be an infinite $\omega$-stable group and $p \in S(G)$ a generic type. What set does $p$ define? Or do we know anything about it? Like where it lies.
I am struggling with generic types/planes/etc in model theory. It would be also great if someone could suggest me some sources about it to read.
 A: In any $\omega$-stable theory $T$, the formula $\top$ or $x = x$ in the context of a single free variable $x$ has a Morley rank $\alpha$ (an ordinal) and a Morley degree $n$ (a natural number). The degree $n$ is the maximal number of pairwise disjoint definable sets of Morley rank $\alpha$. Working in a sufficiently saturated model $M\models T$, $M$ is partitioned into $n$ definable pieces $X_1,\dots,X_n$, each of Morley rank $\alpha$, and this partition is unique up to adjustments by definable sets of rank $<\alpha$. We call the pieces of the partition irreducible components of $M$. 
Now a type $p\in S_1(M)$ is generic if it has Morley rank $\alpha$. Since the $X_i$ form a partition, $p$ must live in in one of the sets $X_i$. But for every definable subset $Y\subseteq X_i$, either $Y$ or $X_i\setminus Y$ has rank $<\alpha$ (otherwise we could split $X_i$ into two pieces of full rank and get a bigger partition). So since $p$ does not live in any definable set of rank $<\alpha$, $p$ is uniquely determined by which irreducible component $X_i$ it lives in. So $S_1(M)$ contains exactly $n$ generic types. 
To your question "where does a generic type lie", the answer is...well, almost everywhere!  (In its irreducible component.) A realization of a generic type $p\in S_1(M)$ (in an elementary extension of $M$) lives in the irreducible component $X$ of $p$, but not in any "small" (in the sense of rank) $M$-definable subset of $X$. So $p$ defines the set of elements of $X$ which do not lie in any lower-rank $M$-definable subset of $X$.  
If you know something about algebraic geometry, a good analogy is that the generic types in an $\omega$-stable theory are like the generic points of irreducible components of an algebraic variety. The Zariski closure of a generic point is its entire irreducible component, because it does not live in any proper Zariski closed subset. 
All of the above was in the general context of $\omega$-stable theories. Specializing to $\omega$-stable groups, we get some additional information about generic types. Let $G$ be a sufficiently saturated $\omega$-stable group.


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*$G$ has a "connected component of the identity" $G^0$, which is the minimal definable subgroup of finite index in $G$. The Morley degree is the index of this subgroup and the irreducible components are its cosets. So there is exactly one generic type living in each coset of $G^0$. 

*A definable subset $X\subseteq G$ has full rank $\alpha$ if and only if finitely many translates of $X$ cover $G$, i.e. $a_1\cdot X \cup \dots \cup a_n\cdot X = {G}$ for some $a_1,\dots,a_n\in {G}$. So a generic type only contains formulas defining "wide" sets in the sense that finitely many translates cover the group. This is a precise sense in which  "where a generic type lives" can't be pinned down very tightly. 

*Any element of $G$ is a product of two generic elements over $G$, i.e. for all $g\in G$ there exist $a,b\in \mathbb{G}$, a sufficiently saturated elementary extension of $G$, such that $a\cdot b = g$ and $\text{tp}(a/G)$ and $\text{tp}(b/G)$ are generic. 
For your reference request: If you are particularly interested in $\omega$-stable theories and $\omega$-stable groups, Marker's book Model Theory: An Introduction gives a good overview (Chapters 6 and 7 - all the facts I wrote above are proven there). $\omega$-stability is a special case of the more general theory of stability. For more information on stable groups in general, the standard reference is Poizat's book Stable Groups. For stable theories in general, there are many references. Which one is best for you depends on your background and interests. 
