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
Andreas Zastrow · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1998
The existence of a group H as described in the title shows that the statement of Rips's Theorem for finitely generated groups cannot be extended without further complications to infinitely generated groups. The construction as given in this paper uses a careful combinatorial description 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.