Euler characteristics for one-relator products of groups

Abstract

We calculate Euler characteristics for one-relator products of groups G = (G(1) * G(2))IN(R-m) 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,C) 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 R-m with a 'multiply exceptional form', and establish a connection with a class of orbifolds studied by Jones and Reid.