CONSTRUCTION OF AN INFINITELY GENERATED GROUP THAT IS NOT A FREE PRODUCT OF SURFACE GROUPS AND ABELIAN-GROUPS, BUT WHICH ACTS FREELY ON AN R-TREE

The existence of a group H as described in the title shows that the st atement of Rips's Theorem for finitely generated groups cannot be exte nded without further complications to infinitely generated groups. The construction as given in this paper uses a careful combinatorial desc ription of the fundamental group of the Hawaiian Earrings and a length function that can be put on a special subgroup. Then the existence of H follows using a theorem of Chiswell, Alperin and Moss.