ABSTRACT
Let d be a positive square-free number. Let Od be the ring of integers in Q ( − d ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2432.tif"/> . The groups PSL (2, Od ) are called collectively the Bianchi groups. The extended Bianchi group Bd is the maximal discrete extension of PSL (2, Od ) in PSL (2, C) (Bianchi [2]). The group Bd acts by linear fractional transformations on the complex plane C. Let RBd be Bd with the generator θ , θ ( z ) : = z ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2433.tif"/> , adjoined. Serre [16] observed that, for small values of d, the groups RBd are the Coxeter groups (i.e. generated by reflections). However, there are only finitely many groups RBd which are Coxeter groups (see [5], [7]) 1 . A Fuchsian subgroup of Bd fixes a circle or straight line C and the two components of C/C. It is non-elementary if its limit set on C has more that two points.
