2009/11/01 by O. Kharlampovich, Kharlampovich, O., A. Myasnikov +3 · 1 citation
Mathematics · #20F65 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20F65
paper · pdf · doi:10.48550/arxiv.0911.0209
33 pages, 6 figures
arxiv created 2011/11/02 · arxiv updated 2015/03/13
A group is called Λ-free if it has a free Lyndon length function in an ordered abelian group Λ, which is equivalent to having a free isometric action on a Λ-tree. A group has a regular free length function in Λ if and only if it has a free isometric action on a Λ-tree so that all branch points belong to the orbit of the base point. In this paper we prove that every finitely presented Λ-free group G can be embedded into a finitely presented group with a regular free length function in Λ so that the length function on G is preserved by the embedding. Next, we prove that every finitely presented group \widetilde G with a regular free Lyndon length function in Λ has a regular free Lyndon length function in \mathbb Rn ordered lexicographically for an appropriate n and can be obtained from a free group by a series of finitely many HNN-extensions in which associated subgroups are maximal abelian and length isomorphic.