A presentation for reduced extended affine Weyl groups

In 1985 K. Saito [Sa1] introduced the concept of an extended affine Weyl gr oup (EAWG), the Weyl group of an extended affine root system (EARS). In [A2 , Section 5], we gave a presentation called "a presentation by conjugation" for the class of EAWGs of index zero, a subclass of EAWGs. In this paper w e will give a presentation which we call a "generalized presentation by con jugation" for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to "a presentation by conjugati on". Our main result states that when the nullity of the EARS is 2, these t wo presentations coincide that is, EAWGs of nullity 2 have "presentation by conjugation". In [ST] another presentation for EAWGs of nullity 2 is given .