We give a presentation of an elliptic Weyl group
W(
R) (=the Weyl group for an elliptic root system
R) in terms of the elliptic Dynkin diagram Γ(
R,
G) for the elliptic root system. The presentation is a generalization of a Coxeter system: the generators are in one to one correspondence with the vertices of the diagram and the relations consist of two groups: i) elliptic Coxeter relations attached to the diagram, and ii) a finiteness condition on the Coxeter transformation attached to the diagram. The group defined only by the elliptic Coxeter relations is isomorphic to the central extension ˜{
W}(
R,
G) of
W(
R) by an infinite cyclic group, called the hyperbolic extension of
W(
R).
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