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!



amalgamated product of groups and representation theory
Let me expand on Mark Sapir's comment with a concrete example. Burger and Mozes famously [constructed a group](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=burger&s5=mozes&s6=products&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=1446574) 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](http://arxiv.org/abs/1210.2691) and the references therein.


