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Euler characteristics for one‐relator products of groups
Author(s) -
Williams Gerald
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm052
Subject(s) - mathematics , euler characteristic , pure mathematics , euler's formula , context (archaeology) , algebra over a field , mathematical analysis , paleontology , biology
Abstract 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.