Question about representation of free products of groups.

Question

1. Does anyone have an idea of books or papers that treats representation theory of free products of groups?

2. What properties of factors of of a free product suggest a possible representation? Examples will help.

Answer 1

The question seems too vague to admit a real answer, but it might be useful to notice that a representation of a free product $G*H$ on a space $V$ is determined by just giving two representations on $V$, one representation of $G$ and one of $H$, with no required connection between the two representations.

Answer 2

If $G$ and $H$ have faithful representations over a field $K$, then $G*H$ has a faithful representation over some extension of $K$. See, for example,

Marciniak, Zbigniew S. A note on free products of linear groups. Proc. Amer. Math. Soc. 94 (1985), no. 1, 46–48.

Summary: For a field $K$, let $\overline{K}$ denote its algebraic closure. Assume that $|\overline{K}:K|=\infty$. Then for any linear groups $G,H \subseteq GL_n(K)$ their free product $G∗H$ can be embedded into $GL_N(K(t))$. Here $N$ is an integer depending on $K$ only and $t$ stands for an indeterminate.