TITLE: amalgamated product of groups and representation theory
QUESTION [6 upvotes]: Let me ask a question which could be quite stupid, but still:
let $G$ be a group which is an amalgamated product of subgroups $A$ and $B$ over $C$:$\; \;$  $G = A \ast_{C} B$ (subgroups are infinite!).
Question:  How representation theory of $G$ is connected to representation theory of $A$ and $B$ (and $C$),  in another words, how $Rep(G)$ is connected to $Rep(A)$ and $Rep(B)$?
Any comments are welcome!

REPLY [6 votes]: Let me expand on Mark Sapir's comment with a concrete example.  Burger and Mozes famously constructed a group of the form
$\Gamma = F_1*_H F_2$
where $F_1,F_2$ are finitely generated free groups, $H$ is of finite index on either side and $\Gamma$ is simple!  Because finitely generated linear groups are residually finite (by a theorem of Mal'cev, say), it follows that $\Gamma$ has no non-trivial finite-dimensional representations.  On the other hand, of course, the free groups $F_i$ have impossibly rich representation theories.
There are some positive results that hold in special cases, such as when amalgamating over cyclic subgroups.  See, for instance, this recent preprint of Jack Button and the references therein.