Euler Characteristics for One-Relator Products of Groups

We calculate Euler characteristics for one-relator products of groups G = (G1 * G2)/N(Rm) under certain conditions on the form of R and the value of m. As special cases, we study one-relator products of cyclics and recover and generalize results of Fine, Rosenberger and Stille. As corollaries to our main results, we give a necessary condition for G to admit a faithful, discrete representation to PSL(2, ?) of finite covolume. In particular, we generalize a result of Hagelberg, Maclachlan and Rosenberger, from the context of generalized triangle groups to that of one-relator products induced by generalized triangle groups. This provides an answer to a question of Fine and Rosenberger. In deriving our Euler characteristic results we study relators Rm with a ?multiply exceptional form?, and establish a connection with a class of orbifolds studied by Jones and Reid.