Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)

We offer a presentation for the Weyl group of an affine reflection system R of type A(1) as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type A(1). Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended affine Weyl group of type A(1). Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type A(1) is the first nontrivial presentation given in this generality, and can be considered as a model for other types.