The Hnn and Generalized Free Product Structure of Certain Linear Groups

Introduction. If d is a positive square-free integer let Id be the ring of integers in Q(\Jd). Id is a Euclidean domain if d = 1, 2, 3, 7, 11. The groups PSL2(/d) = Fd over these Euclidean rings have recently been investigated. Methods for generating presentations as well as actual presentations were given in [2], [3] and [8], while these groups were shown to be describable in terms of generalized free products in [3]. Here we announce several extensions of these results suggested by Karrass and Solitar. We show that the Picard group Tx is decomposable directly as a free product with amalgamated subgroup while the groups r 2 , T7 , Tx t are HNN groups in the sense of [5]. The extensions will be used in [4] to show that these groups are SQ-universal. Throughout we let Fd = PSL2(/d). The cases d = 1, 3. In [9] it was shown that the Picard group F{ contains a subgroup of finite index which is a generalized free product, while in [3], Tj was decomposed as a semidirect product with the subgroup above contained as a subgroup of finite index in the normal factor. Also, in [3], T3 was shown to have a subgroup of finite index which is again a free product with amalgamated subgroup. As a strengthening of these we obtain