Euler characteristics for one‐relator products of groups

We calculate Euler characteristics for one‐relator products of groups G = ( G 1 * G 2 )/ N ( 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.